Fermilab FERMILAB-THESIS-1999-41

CP Asymmetry in the Decay KL → π+π−e+e−

DISSERTATION SUBMITTED TO

THE GRADUATE SCHOOL OF OSAKA UNIVERSITY

IN CANDIDACY FOR THE DEGREE OF

DOCTOR OF SCIENCE

By

Katsumi Senyo

1999

1

Acknowledgments

I first appreciate Prof. Yorikiyo Nagashima, for all research opportunity in HEP field. I was really

enlightened by his expertise. T. Yamanaka gave me a chance to work on indirect CP violation in

a rare K decay experiment, instead of a B physics experiment. I thank him for his supervision

of my study, and really enjoy a chat with him about experimental and theoretical science. Also,

I remember the trans-pacific midnight proofreading of the thesis, especially in the last one week

before the defense.

Other stuffs in Osaka University, J. Haba, M. Takita, M. Hazumi and I. Suzuki gave me very

useful advises and suggestions, in and out of physics. Their humor and wit always relaxed me.

Thankfully, S. Tsuzuki supported my study from Osaka.

My works done at Fermilab, especially the physics analysis, were mostly supported by V. O’Dell.

The discussion with her was really fun. I enjoyed the life on the crossover with L. Bellantoni, P.

Shanahan, M. Pang, and R. Ben-David. I thank their kindness. I also appreciate KTeV people

on the thirteen floor and new muon lab. Of course, I thank all of people working at the KTeV

experiment. Without their efforts, I could not successfully finish my thesis like this.

I feel relaxed with T. Nakaya and K. Hanagaki, and previous generation of graduate students

in Osaka University. Their friendship and useful suggestion encouraged me in any phases of my

analysis. I studied much from them. I thank graduate students in Osaka University for their

cooperation for years.

My parents, Sakae and Kotaro Senyo, patiently waited for my graduation. Their supports

greatly heartened me. I hope they proud of me.

Finally, I thank my wife, Sachiko, and her tenderness. Actually, I enjoyed the life with her in

Chicago. It was very tough time in my life, but we will long for our precious time and good friends

in Fermilab.

2

Contents

1 Introduction 1

1.1 C, P, and CP Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 CP violation in Neutral Kaon System . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Indirect CP Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.2 Direct CP Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.3 Experimental Results of Indirect CP Violation . . . . . . . . . . . . . . . . 5

1.3 KL → π+π−e+e− Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.1 Decay Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.2 Indirect CP violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.3 Direct CP violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.4 gM1 Form Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.5 Experimental Status of KL → π+π−e+e− . . . . . . . . . . . . . . . . . . . 12

1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.1 Purpose of This Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.2 Overview of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Experiment 14

2.1 KL Beam Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1 Beam Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.2 Sweeper and Collimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.3 Vacuum System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.1 Charged Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.2 Trigger hodoscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.3 Electromagnetic Calorimeter . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.4 Photon Vetoes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.5 Hadron/Muon vetoes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.1 Level 1 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.2 Level 2 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

i

2.3.3 Trigger Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.4 Calibration Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4 Data Acquisition System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4.1 Frontend and DAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4.2 Level 3 filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5 Physics Run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Monte Carlo Simulation 35

3.1 Kaon production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 The Decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.1 KL → π+π−e+e− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.2 KL → π+π−π0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.3 Particle Tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.1 Photon Veto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.2 Drift Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.3 CsI calorimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4 Accidentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4.1 Accidental Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4.2 Accidental Overlay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 Event Reconstruction 43

4.1 Track Candidate Finding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1.1 Hit Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1.2 Finding Y track candidates . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.1.3 Finding X track candidates . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2 Cluster Finding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.3 Track-Cluster Matching and Vertex Finding . . . . . . . . . . . . . . . . . . . . . . 45

4.4 Particle Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5 Event Selection 48

5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.2 Basic Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.2.1 Fiducial Volume Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.2.2 Trigger Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.2.3 Consistency Check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.3 Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.3.1 MK Invariant Mass Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.3.2 Total Momentum Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.3.3 Vertex χ2 Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

ii

5.3.4 Mee Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.3.5 p2t Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.3.6 Pp0kine Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.3.7 Optimization of p2t and Pp0kine Cuts . . . . . . . . . . . . . . . . . . . . . 58

5.4 Acceptance and Data After Final Cuts . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.5 Background Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.5.1 Background Level from Each Source . . . . . . . . . . . . . . . . . . . . . . 62

5.5.2 Background estimation with sideband fit . . . . . . . . . . . . . . . . . . . . 63

5.5.3 Summary of Background Level . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.6 KL Flux Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.6.1 Acceptance and Flux Calculation . . . . . . . . . . . . . . . . . . . . . . . . 64

5.6.2 Systematic Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6 Form Factor Measurement 70

6.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.2 Maximum Likelihood Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.3 Data and Monte Carlo Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.4 Error on the Form Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.4.1 Fitting Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.4.2 Smearing by Detector Resolution . . . . . . . . . . . . . . . . . . . . . . . . 79

6.4.3 Vertexing Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.4.4 Drift Chamber Inefficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.4.5 Momentum Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.4.6 Physics Input Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.4.7 Background Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.4.8 Analysis Dependencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.4.9 Systematic Uncertainty: Summary . . . . . . . . . . . . . . . . . . . . . . . 86

6.5 Form Factor Measurement: Summary . . . . . . . . . . . . . . . . . . . . . . . . . 86

7 Asymmetry Measurement 88

7.1 Raw Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.1.1 Raw Asymmetry Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.1.2 Bias from Detector and Event Selection . . . . . . . . . . . . . . . . . . . . 89

7.1.3 Origin of Raw Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7.2 Acceptance Corrected Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.3 Systematic Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

7.3.1 Smearing by Detector Resolution . . . . . . . . . . . . . . . . . . . . . . . . 98

7.3.2 gM1 Form Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.3.3 Vertexing Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

iii

7.3.4 Drift Chamber Inefficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.3.5 Other Physics Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.3.6 Background Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.3.7 Analysis Dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.3.8 Systematic Uncertainty: Summary . . . . . . . . . . . . . . . . . . . . . . . 104

7.4 Asymmetry Measurement: Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 104

8 Branching Ratio Measurement 105

8.1 Event Selection and Background Estimation . . . . . . . . . . . . . . . . . . . . . . 105

8.2 Systematic Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

8.2.1 Drift Chamber Inefficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

8.2.2 Vertexing Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

8.2.3 Normalization Ambiguity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

8.2.4 KL → π+π−e+e− Monte Carlo Generation . . . . . . . . . . . . . . . . . . 110

8.2.5 Background Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

8.2.6 Analysis Dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8.2.7 External Systematic Sources . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8.2.8 Systematic Uncertainty: Summary . . . . . . . . . . . . . . . . . . . . . . . 113

8.3 Branching Ratio Measurement: Summary . . . . . . . . . . . . . . . . . . . . . . . 113

9 Discussion 115

9.1 Form Factor Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

9.2 Asymmetry Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

9.3 Branching Ratio Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

9.4 Remarks and Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

10 Conclusion 119

A Maximum Likelihood Method 121

A.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

A.2 Use of Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

A.3 Reduction of Monte Carlo Generation . . . . . . . . . . . . . . . . . . . . . . . . . 123

A.4 Fit Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

B Pp0kine: Kinematics with

a Missing Particle 125

iv

List of Figures

1.1 KL → π+π−γ decay process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 The photon spectrum in KL → π+π−γ. . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 KL → π+π−e+e− decay process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Critical angles in KL → π+π−e+e−. . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 E∗γ distribution of KL → π+π−γ in the kaon center-of-mass frame. . . . . . . . . . 12

2.1 KTeV beamline and detector layout. . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 KTeV E799-II Detector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 A cross section of the drift chamber. . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 TDC time distribution of drift chamber. . . . . . . . . . . . . . . . . . . . . . . . 20

2.5 SOD distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.6 The V and V’ trigger hodoscope planes. . . . . . . . . . . . . . . . . . . . . . . . . 22

2.7 The KTeV electromagnetic Calorimeter. . . . . . . . . . . . . . . . . . . . . . . . 23

2.8 E/p for electrons in Ke3 decays, where E means an energy deposit in the calorimeter,

and p denotes a momentum measured by the spectrometer. . . . . . . . . . . . . . 25

2.9 The calorimeter intrinsic energy resolution measured in Ke3 events as a function of

electron momentum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.10 The schematic view of RC6, facing downstream. The beam passed through the inner

aperture. The perimeter of outer circle fit right with the vacuum pipe. The other

RCs were basically similar in a form but the dimensions seen in Table 2.2. . . . . . 27

2.11 SA4 from upstream. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.12 The Collar Anti just in front of the calorimeter. . . . . . . . . . . . . . . . . . . . . 27

2.13 The HA hodoscope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.14 The Mu2 counter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.15 Data acquisition system. Three planes were used for Level 3 trigger and the other

was for online monitoring of the detector performance and online calibration. . . . 33

3.1 Momentum distribution of the generated kaons. . . . . . . . . . . . . . . . . . . . . 37

3.2 KL decay Z-distribution of Monte Carlo generated kaons. . . . . . . . . . . . . . . 37

3.3 E/p distribution of hadronic shower samples . . . . . . . . . . . . . . . . . . . . . . 40

3.4 Electromagnetic and hadronic shower comparison. . . . . . . . . . . . . . . . . . . 41

v

4.1 Schematic view of the classification of hit pairs. . . . . . . . . . . . . . . . . . . . . 44

4.2 E/p comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.1 Mass distribution after basic cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.2 Total momentum distribution of four charged tracks. . . . . . . . . . . . . . . . . . 54

5.3 Total momentum from each background source. . . . . . . . . . . . . . . . . . . . . 54

5.4 Vertex χ2 distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.5 Mee distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.6 Mee distribution from each background source. . . . . . . . . . . . . . . . . . . . . 56

5.7 p2t distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.8 pt2 distribution from each background source. . . . . . . . . . . . . . . . . . . . . . 57

5.9 Schematic view of Pp0kine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.10 Pp0kine distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.11 Pp0kine distribution from each background source. . . . . . . . . . . . . . . . . . . 59

5.12 Invariant mass distribution after final cuts. . . . . . . . . . . . . . . . . . . . . . . 61

5.13 Distribution of mass of π+π−e+e−γ. . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.14 HCC threshold distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.1 Mππ distribution between data and Monte Carlo with a constant gM1 in the decay

KL → π+π−e+e−. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.2 The diagrams for the vector meson intermediate model. . . . . . . . . . . . . . . . 72

6.3 The gM1 form factor effect in the decay KL → π+π−γ. . . . . . . . . . . . . . . . 73

6.4 Fit result of a1/a2 and a1 from two dimensional maximum likelihood method. . . . 75

6.5 Comparison of various distributions between data and Monte Carlo. . . . . . . . . 76

6.6 The comparisons of Mππ distribution between data and Monte Carlo with and with-

out the form factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.7 Fit results of a1/a2 and a1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.8 Fit results of a1/a2 and a1 for the ten pseudo data samples. . . . . . . . . . . . . . 80

6.9 Data and Monte Carlo comparison in vertex χ2 for KL → π+π−e+e−. . . . . . . . 81

6.10 Data and modified Monte Carlo comparison in vertex χ2 for KL → π+π−e+e−. . . 81

6.11 Data and Monte Carlo comparison for Mee distribution in the decay KL → π+π−π0D. 83

6.12 Data and Monte Carlo comparison of track illumination at DC1 in Y view. . . . . 84

6.13 Distributions of Mππee from KL → π+π−π0D of data and Monte Carlo. . . . . . . . 85

6.14 Averaged Mππ as a function of (Eπ1 + Eπ2) for KL → π+π−π0D data and Monte

Carlo simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

7.1 φ distribution of data overlaid with the Monte Carlo. . . . . . . . . . . . . . . . . . 90

7.2 Distribution of (2 cosφ sin φ) from data, overlaid with the Monte Carlo. . . . . . . 90

7.3 φ distribution of identified KL → π+π−π0D. . . . . . . . . . . . . . . . . . . . . . . 92

7.4 Raw asymmetries at different mass regions. . . . . . . . . . . . . . . . . . . . . . . 92

vi

7.5 Relation between the input and raw asymmetry. . . . . . . . . . . . . . . . . . . . . 94

7.6 Distributions of KL → π+π−e+e− with respect to φ. . . . . . . . . . . . . . . . . . 96

7.7 Signal acceptance as a function of φ. . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7.8 The φ distribution of acceptance corrected data. . . . . . . . . . . . . . . . . . . . 97

7.9 The resolution effect causing wrong sign events. . . . . . . . . . . . . . . . . . . . . 100

7.10 The difference between real φ and reconstructed φ. . . . . . . . . . . . . . . . . . . 100

7.11 X track swapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.12 The resolution effect causing wrong sign events with Mee < 2MeV/c2. . . . . . . . 102

7.13 The difference between real φ and reconstructed φ for events with Mee < 2MeV/c2. 102

8.1 Invariant mass distribution after final cuts. . . . . . . . . . . . . . . . . . . . . . . 106

8.2 Invariant mass distribution with 0.0001 < pt < 0.0002GeV2/c2. . . . . . . . . . . . 107

8.3 Data and the Monte Carlo comparison after background subtraction. . . . . . . . . 108

8.4 Comparison between data and Monte Carlo estimation in the sideband. . . . . . . 112

9.1 Results on the DE form factor, a1/a2. . . . . . . . . . . . . . . . . . . . . . . . . . 116

9.2 θη−η′ and DE branching ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

vii

List of Tables

2.1 Positions and dimensions of the spectrometer elements. . . . . . . . . . . . . . . . 18

2.2 Positions and dimensions of the photon vetoes. . . . . . . . . . . . . . . . . . . . . 26

2.3 Positions and dimensions of the detectors(downstream of the calorimeter). . . . . . 28

2.4 The E799 four track trigger elements. . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5 The KTeV-E799II run conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1 The parameters referred from Malensek [7] . . . . . . . . . . . . . . . . . . . . . . . 36

5.1 Signal to background ratio matrix table. . . . . . . . . . . . . . . . . . . . . . . . . 60

5.2 The signal acceptance at analysis each phase. . . . . . . . . . . . . . . . . . . . . . 62

5.3 Backgrounds acceptance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.4 Cuts for K0L → π+π−π0

D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.5 Summary of the background level at the flux measurement. . . . . . . . . . . . . . 66

5.6 Summary of systematic uncertainty in the flux measurement. . . . . . . . . . . . . 68

5.7 Summary of the flux calculation with K0L → π+π−π0

D. Expected background was

already subtracted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.1 Fixed physics parameters in the maximum likelihood fit. Values are reported by the

Particle Data Group(PDG) [9] and the theoretical prediction [22]. . . . . . . . . . 74

6.2 Acceptance calculation dependency of the sample size. . . . . . . . . . . . . . . . . 78

6.3 Smearing effect by the detector resolution. . . . . . . . . . . . . . . . . . . . . . . . 79

6.4 Stability to parameter fluctuations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.5 Background uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.6 Deviations of analysis cut dependencies. . . . . . . . . . . . . . . . . . . . . . . . . 87

6.7 Systematic uncertainties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

7.1 Raw asymmetries at different mass regions. . . . . . . . . . . . . . . . . . . . . . . 91

7.2 Comparison between the input and raw asymmetry. . . . . . . . . . . . . . . . . . . 93

7.3 Asymmetries of various data sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

7.4 Experimental Input Parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.5 Deviations of analysis cut dependencies in the acceptance corrected asymmetry. . . 103

viii

7.6 Summary of systematic uncertainties. . . . . . . . . . . . . . . . . . . . . . . . . . 104

8.1 Background acceptance after final cut. . . . . . . . . . . . . . . . . . . . . . . . . . 107

8.2 The signal acceptance at each analysis phase. . . . . . . . . . . . . . . . . . . . . . 108

8.3 Stability to parameter fluctuations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

8.4 Summary of analysis dependency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8.5 Summary of uncertainty in branching ratio measurement. . . . . . . . . . . . . . . 114

ix

Abstract

CP violation in an angluar asymmetry has been established in the decay KL → π+π−e+e− based on

1162 events. Measured CP asymmetry, which appeared in the angular distribution between normals

to π+π− and e+e− planes, is 0.127 ± 0.029(stat.) ± 0.016(syst.). The form factor of M1 direct

emission appeared in KL → π+π−γ∗(→ e+e−) is determined as a1/a2 = −0.684+0.031−0.043(stat.) ±

0.053(syst.) and a1 = 1.05±0.14(stat.)±0.18(syst.). The branching ratio has also been determined

as Br(KL → π+π−e+e−) = (3.55± 0.11(stat.)± 0.08(syst.internal) ± 0.14(syst.external)) × 10−7.

Chapter 1

Introduction

The CP violation is a small effect, yet it has been one of the greatest discoveries in particle physics.

Since its discovery in 1964, people have been eager for elucidating its origin, process and effect.

Recently, we are finally recognizing its phenomenology in the electroweak physics. This thesis

intends to show a part of such challenging efforts by exploring the decay “KL → π+π−e+e−.”

In this introduction, we cover theoretical backgrounds of KL → π+π−e+e− decay process. It

starts with a historical overview on C, P and CP violations. Next, we describe phenomenology of

the neutral kaon system and CP violation. In addition, we briefly introduce recent experimental

status of its direct and indirect CP violation effects. We then study the decay KL → π+π−e+e−

process and the technique to measure the CP violation effect in this decay mode. Finally, we glance

the overview of this thesis.

1.1 C, P, and CP Violation

Most of the theories in particle physics declare the existence of some kinds of symmetries in the

framework. However, observations of symmetry breakings in nature have been a guide to a new

theory, which declares underlying symmetry in the more fundamental level.

In particle physics, there are three interesting discrete symmetries; charge conjugation(C),

parity, i.e., space inversion(P), and time reversal(T). These symmetry operations have eigenvalues

of +1 and -1, called even and odd, respectively. Relativistic field theories require that the equations

of motion must conserve its form under joint operations of C, P, and T. This is called a CPT

theorem [1, 2]. The CPT theorem assures the equality of masses, lifetimes and magnitude of charges

between particles and anti-particles. To date, no CPT violation has been observed experimentally.

At the dawn of modern particle physics, any physical processes were simply believed to be

invariant under each operation of C, P, and T. In the early 50’s, strange particles were found.

Among them were those named “τ” and “θ.” Although they had the same lifetime and mass, these

two particles decayed into different parity states, actually two pions and three pions respectively,

1

so that they were considered as different particles. In 1956, Lee and Yang insisted that no obvious

evidence for the parity conservation had been found in the weak interaction [3]. They proposed

experiments to test parity conservation in β decays of a polarized nuclei and so on. An experiment

to test the invariance was performed in 1957 by Wu et al.[4], using the Gamow-Teller transition in60Co polarized nuclei, and they discovered the parity violation in the decay. With further studies

on the weak interaction, it was found that C and P symmetries were fully violated, but still CP

was thought to be invariant at that time.

In 1964, even CP symmetry was found to have a small but finite amount of violation in the

long-lived neutral kaon system [5]. The long-lived kaon, KL, which had been believed to be a pure

CP odd state, was found to decay into CP even two pion state. Up to date, C, P and CP violations

are known in the weak interaction only, and especially, CP violation has been observed only in the

neutral kaon system.

Since the discovery of the CP violation, people have been exploring its mechanism and origin,

and recently with sophisticated detectors and techniques, we have accumulated enough knowledge

to almost clarify the nature of CP violation in the electroweak interaction.

1.2 CP violation in Neutral Kaon System

In this section, we explain phenomenological description of CP violation in the neutral kaon system

and recent experimental status of them. First, we start with the phenomenology.

There are two kinds of neutral kaons in different eigenstates which are distinguished with

strangeness of +1 and -1:

|K0 >= |(d, s) >, |K0 >= |(d, s) > . (1.1)

They are easily produced through the strong interaction, where the strangeness is conserved.

K0 and K0 are defined as CP conjugate to each other:

CP |K0 > = |K0 >, (1.2-a)

CP |K0 > = |K0 >, (1.2-b)

with phase ambiguity, where CP means the CP transforming operation. Here we can choose the

phase as above since the phase is not observable. If we define K1 and K2 as

|K1 > =1√2(|K0 > +|K0 >), (1.3-a)

|K2 > =1√2(|K0 > −|K0 >), (1.3-b)

these are CP eigenstates like

CP |K1 > = |K1 > (CP even), (1.4-a)

CP |K2 > = −|K2 > (CP odd). (1.4-b)

2

K0 and K0 can transit to each other through K0 ↔ (2π, 3π etc.) ↔ K0, implying < K0|H |K0 >6=0, where H includes the strong interaction HS , and the weak interaction HW causing decays. This

phenomenon is called “mixing.” Naturally, the mixing includes ∆S = 2 transition.

To understand the mixing in the neutral kaon system in the rest frame, let us start from a

time-dependent Schrodinger equation with two quantum states including the mixing like [6];

φ(t) = α(t)|K0 > +β(t)|K0 >,

id

dt

(α

β

)=

(< K0|H |K0 > < K0|H |K0 >

< K0|H |K0 > < K0|H |K0 >

)(α

β

)

=

(M11 M12

M21 M22

)(α

β

). (1.5)

Taking the second order of the weak interaction, we can rewrite Mij as a combination of two

Hermitian matrices, Mij and Γij ;

Mij = Mij − iΓij/2 (1.6-a)

Mij = Miδij+ < i|HW |j > +∑

n 6=K0,K0

P< i|HW |n >< n|HW |j >

Mi − En(1.6-b)

Γij = 2π∑

n 6=K0,K0

δ(Mi − En) < i|HW |n >< n|HW |j >, (1.6-c)

where P means a principal value. The M is called mass matrix and Γ is called decay matrix, and

its off-diagonal elements give the mixing between K0 and K0 states.

We now consider the eigenvalue of this equation. We can rewrite

φ(t) = α′(t)|K1 > +β′(t)|K2 >= α(t)|K0 > +β(t)|K0 > (1.7-a)

id

dt

(α′

β′

)=

(λS 0

0 λL

)(α′

β′

)(1.7-b)

λS = mS − iΓS/2, λL = mL − iΓL/2, (1.7-c)

where the eigenvalues are, in general, complex and mL,S , ΓL,S are real.

Since only K1 can decay into ππ*1, the Q value of K1 is larger than that of K2. Therefore, K2

lives longer than K1(Γ1 Γ2). Until CP violation was observed in 1964, it was believed that the

long lived kaon could have decayed only into odd CP state.

*1Here is an explanation of CP state in 2π and 3π system:

1) 2π: Pπ = −π and Cπ0 = π0, and Cπ± = π∓. Because of Bose statistics, C is equivalent to P in π+π− system.

So CP (π+π−) = (−1)2` = +1, where ` is an orbit angular momentum. For 2π0, an orbit angular momentum must

be an even number from the Bose symmetry, thus CPπ0π0 = +1.

2) 3π: For π+π−π0, assuming the angular momentum of π+π− is ` and a relative angular momentum between a

π0 and π+π− system is L, it gives CP (π+π−π0) = (−1)2`+L+1. Since the angular momentum barrier forces to be

L = 0 and ` = 0, CP becomes negative. For 3π0, like 1), ` becomes zero. So L = 0 since the total momentum must

be zero. Thus, CP (π0π0π0) must be negative.

3

1.2.1 Indirect CP Violation

CP violation was first observed as a signature of long lived kaons decaying into CP even two pion

final states. This phenomenon is explained if the long lived kaon, KL, is almost K2, but has an

admixture of CP even K1 state.

|KL >=1√

1 + |ε|2 (|K2 > +ε|K1 >) =1√

2(1 + |ε|2) ((1 + ε)|K0 > −(1 − ε)|K0 >), (1.8-a)

|KS >=1√

1 + |ε|2 (|K1 > +ε|K2 >) =1√

2(1 + |ε|2) ((1 + ε)|K0 > +(1 − ε)|K0 >), (1.8-b)

where we assume CPT invariance. The contamination of K1 in the KL causes the decay into CP

even final state, such as two pions, and amount of CP asymmetry is determined by ε. The CP

violation from the mixing asymmetry is called “indirect CP violation.”

1.2.2 Direct CP Violation

The CP violation can be observed in a decay process also. This is called “direct CP violation.” If

we define decay amplitudes to a CP eigenstate fCP like

afCP = < fCP |Heff |K0 >

afCP = < fCP |Heff |K0 >, (1.9)

direct CP violation means afCP 6= afCP . Let us choose CP even final state, fCP , and define a CP

violating parameter:

rfCP =< fCP |Heff |KL >

< fCP |Heff |KS >. (1.10)

Substituting Equation 1.9 into Equation 1.10, we obtain

rfCP =(afCP − afCP ) + ε(afCP + afCP )(afCP + afCP ) + ε(afCP − afCP )

. (1.11)

If afCP 6= afCP , not only ε term(i.e., indirect CP violation) but also the difference between afCP

and afCP contribute to the CP violation. Defining the decay asymmetry as

χfCP =afCP − ¯afCP

afCP + ¯afCP

, (1.12)

we can rewrite Equation 1.10 as

rfCP =ε + χfCP

1 + εχfCP

≈ ε + χfCP (1.13)

where we used an empirical fact that ε is very small, ∼ 10−3. Equation 1.13 shows CP violation

can be separated into two effects.

The direct CP violation was predicted by many people. Among them, Kobayashi and Maskawa

showed that CP violation results by extending the quark mixing from four quarks to six quarks [33],

4

which is now a part of the Standard Model. Since then, many searches have been performed, and

finally, a finite direct CP violation has been reported by NA31 and KTeV collaborations in the

neutral kaon system [34, 35].

1.2.3 Experimental Results of Indirect CP Violation

We briefly introduce experimental results in the field of indirect CP violation. We here show

results from KL → ππ, a semi-leptonic K decay, and a radiative K decay, KL → π+π−γ. The

decay KL → π+π−γ is strongly related to the decay KL → π+π−e+e−, which is the topic of this

thesis.

Indirect CP violation in KL → ππ

CP violation was first observed in KL → π+π− by V.L.Fitch, J.W.Cronin et al. in 1964[5]. This

was a direct observation of indirect CP violation with CP even final state, π+π−. This implies

that KL includes the CP even small component of K1 which decays into π+π− final state.

Now the branching ratio is known to be [9]

BR(KL → π+π−) = (2.067± 0.035)× 10−3. (1.14)

CP violation was also observed in a final state, KL → π0π0,

BR(KL → π0π0) = (9.36± 0.20) × 10−4. (1.15)

The related useful CP violation parameters are also provided from the decay amplitudes:

η+− ≡ Amp(KL → π+π−)Amp(KS → π+π−)

= |η+−| exp(iΦ+−),

η00 ≡ Amp(KL → π0π0)Amp(KS → π0π0)

= |η00| exp(iΦ00).

Recent results of these parameters are [9]

|η+−| = (2.285± 0.019)× 10−3, Φ+− = 43.5 ± 0.6,

|η00| = (2.275± 0.019)× 10−3, Φ00 = 43.4 ± 1.0.

Experimentally, a simultaneous collection of KL → π+π−, π0π0 and KS → π+π−, π0π0 decays

gives more precise measurements of the ratio, |η00/η+−|, and the phase difference, ∆Φ ≡ Φ00 −Φ+− [9]:

| η00

η+−| = 0.9956± 0.0023, ∆Φ = −0.1 ± 0.8.

5

Indirect CP violation in KL → π±`∓ν

Another place to observe the CP violating effect is the charge asymmetry in kaon semileptonic

decay, KL → π±`∓ν. The charge asymmetry is defined as

δ ≡ Γ(KL → π−`+ν`) − Γ(KL → π+`−ν`)Γ(KL → π−`+ν`) + Γ(KL → π+`−ν`)

. (1.16)

These decay amplitudes indicate |K0 > and |K0 > composition of the KL, since an `+ can come

only from a K0 decay, while `− can come only from K0 decay, as long as the “∆S = ∆Q rule”

holds. CPLEAR experiment at CERN reported that a deviation from ∆S = ∆Q rule in K decay is

consistent with zero [10]. Therefore, the charge asymmetry is calculated from relative amplitudes

of 1 + ε for K0 → π−`+ν`, and 1 − ε for K0 → π+`−νe:

δ =|1 + ε|2 − |1 − ε|2|1 + ε|2 + |1 − ε|2 ≈ 2Re(ε). (1.17)

Averaging the results from KL → π±e∓ν(called “Ke3”), and KL → π±µ∓ν(called “Kµ3”),

the experimental result was reported [9]:

δ = (3.27± 0.12) × 10−3. (1.18)

Using the phase of ε, this corresponds to |ε| = (2.25± 0.09) × 10−3.

These facts indicate that the same mixing mechanism is(at least) mostly responsible for CP

violation effects in KL → ππ and the semileptonic decays.

CP violation in KL → π+π−γ

The decay process of KL → π+π−γ consists of (a) an inner bremsstrahlung(IB) component as-

sociated with the CP violating decay, KL → π+π−, and (b) an M1 direct photon emission(DE)

component of a CP conserving magnetic dipole moment [29](Figure 1.1)*2.

The simultaneous existence of IB and DE amplitudes implies that the final state contains both

CP = +1 and −1 states.

The radiative photon spectrum from this decay mode becomes a superposition of these two

amplitudes. The photon spectrum of KS,L → π+π−γ in the kaon center-of-mass system measured

by Fermilab E731 experiment is shown in Figure 1.2 [37]. The dotted line is obtained from the

KS → π+π−γ photon spectrum, which is completely dominated by IB component. The data

points indicate the Eγ spectrum for the KL data. For the KL decay, there is an enhancement in

the higher photon energy region over KS → π+π−γ. This is naturally understood to be the DE

component in the KL decay.

Experimental results [36, 37, 38] on the branching ratio of KS → π+π−γ is

Γ(KS → π+π−γ, E∗γ > 20MeV)

Γ(KS → π+π−)= (7.10± 0.16)× 10−3(IB), (1.19)

*2General formalism on radiative K decays is found in Ref. [20, 21]. In this thesis, in accordance with Ref. [29],

we take the highest order of the magnetic dipole expansion, M1, which corresponds to a final state with a CP

eigenstate, CP = (−1)1 = −1. Note that the bremsstrahlung amplitude only includes CP even states.

6

KL

π+

γ

π−

a) Bremsstrahlung

KL

π+

γ

π−

b) M1 Direct Photon Emission

Figure 1.1: Major KL → π+π−γ decay process. (a) an inner bremsstrahlung(IB) component

associated with the CP violating decay, KL → π+π−, and (b) an M1 direct photon emission(DE)

component of a CP conserving magnetic dipole moment.

Figure 1.2: The photon spectrum in KL → π+π−γ [37]. The data points indicate the Eγ spectrum

for the KL data. The dotted line is the same spectrum for the KS data, rescaled to the KL data.

7

corresponding to the IB term. The QED prediction of this ratio is 7.01 × 10−3 [39], in good

agreement with the experimental result. The KL data was fitted with IB component from KS

data and a pure DE spectrum based on Ref. [29]:

Γ(KL → π+π−γ, E∗γ > 20MeV)

Γ(KL → π+π−)= (7.31± 0.38) × 10−3(IB) (1.20-a)

+(15.7± 0.7) × 10−3(DE) [37]. (1.20-b)

This indicates the existence of two components in the KL decay.

1.3 KL → π+π−e+e− Decay

This section describes a theoretical and experimental interpretation for the decay KL → π+π−e+e−

from an aspect of CP violation.

In the previous section, we looked at theoretical and experimental results of KL → π+π−γ. If

we consider a virtual transition of the photon, γ∗ → e+e−, another CP violating effect will appear

in this decay. Here, we explain how CP violating effect arises in this decay.

1.3.1 Decay Processes

We start this section with a brief introduction to the theoretical overview of the physics process

in KL → π+π−e+e−. We first define decay amplitudes contributing to this decay. Similarly

to the decay KL → π+π−γ, the most dominant contribution is the M1 magnetic dipole direct

emission(DE) and the inner bremsstrahlung photon emission(IB).

The dominant decay amplitude of

KL(P) → π+(p+)π−(p−)e+(k+)e−(k−) (1.21)

has the form

M(KL → π+π−e+e−) = Mbr + Mmag, (1.22)

where

Mbr = e|fS|gBR[p+µ

p+ · k − p−µ

p− · k ]e

k2µ(k−)γµµ(k+), (1.23-a)

Mmag = e|fS|gM1

M4K

εµνρσkνpρ+pσ

−e

k2µ(k−)γµµ(k+). (1.23-b)

The terms Mbr and Mmag denote the bremsstrahlung and magnetic dipole, respectively. The

coefficients appearing there are

gBR = η+−eiδ0 ,

gM1 = i(0.76)eiδ1 ,

8

and |fS | is defined by

Γ(KS → π+π−) =|fs|2

16πmK[1 − 4m2

π

m2K

]. (1.25)

The η± is defined as the ratio of amplitudes Amp(KL → π+π−)/Amp(KS → π+π−) as shown

in Section 1.2.3, and δ0(1) is a phase shift in the final-state interactions in the ππ system with

I = 0(1) wave state.

KL

π+

e+γ∗

e-

π−

a) Bremsstrahlung

KL

π+

e+γ∗

e-

π−

b) M1 Direct Photon Emission

KL

π+

e+γ∗

e-

π−

c) E1 Direct Photon Emission

KL

π+

e+γ∗

e-

π−

d) K0 Charge Radius

KS

KL

π+

e+

e-

π−

e) SD Direct CP Violation

Figure 1.3: KL → π+π−e+e− decay process. Besides major contributions of a) Inner

Bremsstrahlung and b) M1 direct emission, there are other small contributions.

In addition, there are three small contributions; E1 electro dipole direct emission, charge radius

and short distance direct CP violating amplitudes.

By integrating over all parameters, the branching ratio is calculated as

Br(KL → π+π−e+e−) ≈ 3 × 10−7 [22, 23]. (1.26)

1.3.2 Indirect CP violation

Here we define critical parameters in KL → π+π−e+e−. This decay kinematics is defined com-

pletely with five kinematical parameters; invariant mass of π+π−, invariant mass of e+e−, an angle

θπ+ between the directions of electrons and π+ in the pions center-of-mass frame, and an angle

9

θe+ between the directions of pions and e+ in the electrons center-of-mass frame. The angle φ,

which is of most interest in this thesis, is defined in terms of unit vectors constructed from the

pion momenta p± and lepton momenta k± in the KL rest frame:

nπ = (p+ × p−)/|p+ × p−|, n` = (k+ × k−)/|k+ × k−|,z = (p+ + p−)/|p+ + p−|,

sinφ = nπ × n` · z, cosφ = nπ · n`. (1.27)

Thus, the angle φ is simply understood as an angle defined between normals to π+π− and e+e−

decay planes in the KL rest frame. The schematic view of these angles is shown in Figure 1.4.

π+

e-

π-

e+

φKL

KL

π+

π-

e+ e-P

θπ+

KL

π+π-Pθe+

e+

e-

φ is an angle between the normals to the ee and ππ planes in Kaon center of mass frame.

θπ+ is an angle between a center of mass of two electrons and π+ in pions' center of mass frame.

θe+ is an angle between a center of mass of two pions and e+ in electrons' center of mass frame.

Figure 1.4: Critical angles in KL → π+π−e+e−.

Integrating over all kinematic variables except for φ, and ignoring small contributions, the

differential cross section in terms of these angles is written as follows;

dΓdφ

= Γ1 cos2 φ + Γ2 sin2 φ + Γ3 sin φ cosφ. (1.28)

To identify the CP-violating terms in this expansion, we write the terms under the CP transfor-

mation, p± → −p∓, k± → −k∓, which gives;

cosφ → + cosφ,

sin φ → − sinφ. (1.29)

10

Those constants were evaluated numerically in Ref. [23]. The Γ3 is the CP violating contribution,

which is caused by the interference between IB and DE terms. In this sense, this is indirect CP

violation. This contribution gives the angular asymmetry of

Asym. =

∫ π/2

0dΓdφdφ − ∫ π

π/2dΓdφdφ∫ π/2

0dΓdφdφ +

∫ π

π/2dΓdφdφ

≈ 14% [22, 23], (1.30)

where we folded the distribution within −2π < φ < 0 over 0 < φ < 2π for simplicity. This suggests

that there is a large CP violating effect in this decay mode.

1.3.3 Direct CP violation

The angle φ is related to indirect CP violation, while θπ+ and θe+ is expected to exhibit direct CP

violation by interference between short-distance direct CP violating term and other contributions

at very small level [23]. The theory suggests that the interference can be at most of order 10−6 as

compared with the DE contribution. In addition, the direct CP violating effect does not contribute

to the angular asymmetry of φ, shown in Equation 1.30. Therefore, the direct CP violating effect

can be neglected in this study.

1.3.4 gM1 Form Factor

The theoretical calculations [22, 23] for the angular asymmetry and branching ratio of KL →π+π−e+e− used a constant form factor gM1 for the M1 direct emission contribution, which was

taken from the experimental result of KL → π+π−γ [22, 37]. However, an effect of vector meson

intermediate contribution was predicted [29] and observed [37] in the parent process, KL → π+π−γ.

The form factor gM1 is modified as;

gM1 −→ gM1 · FF = a1[(M2

ρ − M2K) + 2MKEγ ]−1 + a2, (1.31)

where Mρ is the mass of ρ vector meson, MK is the mass of kaon and Eγ is the energy of the

photon in the KL rest frame. Figure 1.5 shows distributions of Eγ with vector meson intermediate

contribution or a constant gM1 [37]. Clearly, inclusion of the vector meson effect in the form

factor gives a better agreement.

Besides KL → π+π−γ, the form factor gM1 can be independently measured with KL →π+π−e+e−, in principle. The measurement should improve the estimate of the branching ratio

and CP violating asymmetry of KL → π+π−e+e−. From a point of view of our experiment, the

uncertainty in gM1 gave an uncertainty in the branching ratio measurement up to 10%, while the

expected statistical uncertainty would be 3%. The accuracy of the branching ratio measurement

would be systematic dominant unless we improve the accuracy of gM1 form factor.

11

Figure 1.5: E∗γ distribution of KL → π+π−γ in the kaon center-of-mass frame [37]. Dots are from

data and dotted histograms are from Monte Carlo simulation. a)Monte Carlo with a form factor

taking account of vector meson intermediate. b)Monte Carlo with a constant gM1.

1.3.5 Experimental Status of KL → π+π−e+e−

The decay KL → π+π−e+e− was first observed in 1998 by the KTeV collaboration with 46

events [40] based on one day of data. The branching ratio was reported as:

Br(KL → π+π−e+e−) = (3.2 ± 0.6(stat.) ± 0.4(syst.)) × 10−7, (1.32)

and this result agrees well with theoretical predictions, 3 × 10−7. However, statistics was not

enough to evaluate the CP violating effect as well as the M1 direct emission form factor.

1.4 Summary

1.4.1 Purpose of This Study

The decay KL → π+π−e+e− is a fruitful field to explore CP violation in particle physics. This

mode is expected to show not only the fourth appearance of indirect CP violation, but also the

significant CP violating effect.

In addition, we can measure the form factor of M1 direct emission contribution in KL →π+π−e+e− for the first time. This serves as a cross check of the form factor measured with

KL → π+π−γ.

12

Therefore, we will extract the following parameters of KL → π+π−e+e− in this thesis:

• gM1 direct emission form factor.

• CP asymmetry in the angular distribution.

• Branching ratio.

1.4.2 Overview of This Thesis

We have discussed a theoretical aspect of KL → π+π−e+e−, especially in CP violation and the

form factor of M1 direct emission in this chapter. The rest of this thesis is devoted to describe

new experimental results on KL → π+π−e+e− in the KTeV experiment. In the following chapters,

we start from explaining the detectors and run conditions. To understand detector response and

acceptance, we utilized a Monte Carlo simulation, described in Chapter 3. Event reconstruction

strategy and its scheme in the analysis are explained in Chapter 4. After that, we describe event

selection to extract the signal.

We proceed to physics analyses after the signal is obtained. First, we determine the form factor

of M1 direct emission, and then feedback the measured parameters to the Monte Carlo simulation

to improve its accuracy. Next, the angular asymmetry from the CP violating effect is evaluated

and then, the branching ratio of this mode is determined. Finally, we summarize and discuss our

results.

13

Chapter 2

Experiment

This analysis on KL → π+π−e+e− was performed as a part of a KL rare decay search program

KTeV–E799II in the Fermi National Accelerator Laboratory in 1997.

This chapter describes the KTeV experiment and consists of three parts; short explanation of

the data collecting strategy of KL → π+π−e+e−, KLbeam and detector configuration, and data

taking. We briefly mention the issues only related to the KL → π+π−e+e− analysis. More detailed

information about the KTeV experiment can be found in KTeV Technical Design Report [8] and

KTeV internal documents.

The final state of the decay KL → π+π−e+e− consists of four charged tracks, π+π−e+e−. This

means if all four particles are identified and momenta are measured well, the whole kinematics and

decay topology can be fully reconstructed.

The decay KL → π+π−e+e− was normalized by KL → π+π−π0 with π0 Dalitz decay(denoting

“π0D”), π0 → e+e−γ. In order to identify this decay, we should also detect a photon and measure

the photon energy and position. Since this decay has a final state of π+π−e+e−γ, this can be a

significant background if one photon is missed. Therefore, photon vetoes were also required to veto

events with escaping particles.

From this regard, detector elements are required to have following functions:

• Momentum measurement of charged particles

• Identification of e±, π± and µ±.

• Photon energy and position measurement.

• Vetoes for the particles escaping from the detector.

The KTeV detector was capable of satisfying these requirements. We here describe the KTeV

detector system in this context.

14

Before proceeding to the description of the experiment, the coordinate system of the KTeV

experiment is defined here. A target is the origin of the coordinate, and downstream horizontal

direction along beams is defined as the positive Z axis. The Y axis is defined as vertically pointing

up. The three axes are defined with the right-handed coordinate system.

2.1 KL Beam Production

This section describes the neutral beam production.

A 800 GeV primary proton beam was incident on a BeO target, to create various particles.

Among them, neutral particles were selected by series of dipole magnets and collimators. The

beam was then transported to the experimental hall.

2.1.1 Beam Production

Tevatron main ring in Fermilab provided 800GeV protons to KTeV. Protons were delivered over

a 23 seconds “spill” in every 60 seconds. The intensity of the proton beam provided to KTeV

was 2.5× 1012 to 5.0× 1012 protons per spill. Each spill has microstructure divided with 53 MHz

Tevatron radio frequency(RF), that is, a few nano second wide proton bunch delivered in every 19

nsec.

The primary proton beam was incident on the target with the vertical angle of 4.8 mrad in

order to optimize both the higher neutral kaon flux and the better kaon to neutron flux ratio. The

target was made of beryllium oxide(BeO) with the dimensions of 3×3 mm transverse to the proton

beam and 30 cm(0.9 interaction lengths) along the beam.

2.1.2 Sweeper and Collimeter

A sweeper magnet(NM2S1), which was located two meters downstream from the target, removed

the remaining primary protons from the neutral beam. The primary proton beam was absorbed

by a beam dump.

Another three dipole magnets were placed in the region of Z = 14 to 93m from the target, to

remove charged particles from the beam, including upstream decay, muons from the beam dump,

and interaction products.

A Pb filter(NM2PB) at Z = 18.5 m converted photons in the neutral beam into e+e− to remove

them effectively.

Two nearly parallel neutral beams were defined with a primary collimeter with two square holes

placed at Z = 19.8 meters. A steel slab collimeter at Z = 38.8m prevented particles crossing from

one beam to the other.

At Z = 85 m, a collimeter named “the defining collimeter”, which was made of steel, defined

the final dimensions of the beams. The beams formed square dimensions of 0.5 mrad × 0.5 mrad

for the first part of the run(winter run) and 0.7 mrad × 0.7 mrad for the second part of the

15

NM2

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Version 5.0

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Legend

Walls of existing structures

Walls of new construction

Deep Monuments

Sight Risers

~ Loss monitors

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KTeV Beamline & Experimental Hall Layout

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(Conceptual)

Mask-anti(E832)

Defining Collimator

Slab Collimator(NM2CV1)

BA

1.8 m vacuum window

HeHe

AVB B2 dipolesFinal Focus Quads

Final Sweeper

Active Regenerator (E832)

E799/E832 Experimental Hall

Decay RegionHadron

Veto(µ1)

µ2

µ3

CASA4

AAAAAAAAAAAAAAAAA

Moveable Absorber (Be)(NM2ABS1)

Primary Collimator(NM2TCOLL)

µsweep1(NM2S2)

µsweep2(NM2S3)

Primary Proton Dump(NM2BD)

µ Filter &BeamDump

µ F

ilter

CsI Calorimeter

BlockHouse

He He

AAA

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TriggerHodo.

Jaw Collimators

18" decay pipe 30" decay pipe

Spin Rotator Dipole

00

meters

feet

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60

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KAMI Muon Range Stack

0

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70 80 90 100 110 120 130 140 150 160 170 180 190 200

meters

feet

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300 500 600400

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meters

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KTeV Pre-target AreaKTeV Target Hall

KTeV Secondary Neutral Kaon Beam

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Beamstop(NM2BS)

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30 cm BeO Target

Eartly Sweeper (NM2S1)

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(Air SWIC)

(NM2TGT)

NM2BPMV2NM2BPMH2

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~~ ~ ~ ~ ~ E8 HyperonMagnet

Common Absorber (Be)(NM2ABS2)

Pb Absorber(NM2PB)NM2USP

104,060DUSAF

Figure

2.1:K

TeV

beamline

anddetector

layout.

16

run(summer run), and their centers were separated by 1.6 mrad. The slab collimeter was removed

in the summer run.

At 90 m from the target, downstream of the defining collimator, there was a final sweeper

magnet to remove charged particles produced in upstream decays or interaction. This provided a

transverse momentum of 1.1 GeV/c. At this point, the beam mostly consisted of neutrons and KL.

The ratio of neutron and K was approximately 3 and a small number of KS , Λ, and Ξ remained

in the beam.

2.1.3 Vacuum System

A vacuum pipe was located from Z = 22 to 159 meter and its inside was evacuated in order to

reduce beam interaction with residual gas. The vacuum was kept at 1.0× 10−6 torr.

The downstream end of the vacuum pipe at Z = 159 m was sealed with a window, which was

made of Kevlar web and aluminum mylar corresponding to 0.16% radiation lengths and diameter

of 0.9 m.

2.2 Detector

This section describes functions of detector elements used in the KTeV experiment. This includes

explanations of a spectrometer, a electromagnetic calorimeter, and veto system.

The momentum of a charged particle was measured by a spectrometer which consisted of four

drift chambers and a dipole magnet. The energy of a photon was determined with a calorimeter

made of pure CsI crystals. The information of the track momentum and deposit energy of a charged

particle was also used to discriminate a pion and a electron. Photons escaping from the detector

was vetoed by photon veto counters placed at perimeters of the vacuum pipe, drift chambers, and

the CsI calorimeter.

2.2.1 Charged Spectrometer

The KTeV charged spectrometer was located just downstream of the vacuum window. The main

purpose of the spectrometer was to find trajectories of charged particles including the decay vertex

and to measure the momentum of each charged track. In addition, the upstream chambers were

used to provide fast trigger signals.

The spectrometer consisted of four drift chambers(DC1 – DC4) and an analysis magnet placed

between DC2 and DC3. Positions and dimensions of each drift chamber and the analysis magnet

are shown in Table 2.1. The position resolution of the drift chambers was about 100µm and a

nominal momentum resolution was 0.5 %. Helium bags filled between the drift chambers to reduce

the effect of multiple scattering, photon conversion and beam interaction in the spectrometer.

17

20 cm

100 120 140 160 180Distance from Target (m)

TriggerHodoscopes

2 KL beams

TRD

Vacuum Decay Region

Vacuum Window

Analysis Magnet

DriftChambers

MuonCounters

MuonFilter

Hadron Vetowith Lead Wall

CsIPhoton Veto Detectors

Figure 2.2: KTeV E799-II Detector. This is a schematic view of the detector region, as a down-

stream part of Figure 2.1. Note the scale of Z direction, corresponding the position from the target,

is shrunk.

Table 2.1: Positions and dimensions of the spectrometer elements. Z positions are measured at

upstream faces.

Elements Z position(m) Aperture(width(m) × hight(m))

DC1 159.42 1.30× 1.30

DC2 165.57 1.64× 1.44

Magnet 170.01 2.90× 2.00

DC3 174.59 1.74× 1.64

DC4 180.49 1.90× 1.90

18

Drift Chamber

Figure 2.3 shows the cross section of each drift chamber. Each chamber had four sense plane for

the detection of signals from passing charged tracks. A upstream set of two sense wire planes in a

chamber were strung vertically and used to detect X position(X view) and downstream sense wire

planes were for the detection of Y position(Y view).

12.7mm

(0.1mm gold-plated aluminum)Window Guard Wire

Sense Wire(0.025mm gold-plated tungsten)

(0.1mm gold-plated aluminum)Field Wire

Chamber window

Chamber window

Beam Direction

t1, x1

t2, x2

Figure 2.3: A cross section of the drift chamber. This is a look-down view of the drift chamber.

The wires oriented vertically in the front were used for the X position measurement, and the rear

part strung horizontally was for the Y position measurement.

The sense wires were 0.025 mm thick gold-plated tungsten and the field wires were 0.1 mm 6 ogold-plated aluminum. The field shaping wires were arranged in a hexagonal pattern around each

sense wire and formed a hexagonal drift cell. The window guard wires were also 0.1 mm gold-plated

aluminum, to maintain the field shape at the edge of drift cells. The sense wires in each sense plane

were strung with a spacing of 12.7 mm(a half inch), and the adjacent sense planes were staggered

by 6.35 mm with respect to the sense wire positions. This offset allowed us to resolve the left-right

ambiguity of a particle.

The chambers were filled with Argon/Ethane gas mixture. The gas was a 49.75% Argon,

49.75% Ethane and 0.5% isopropyl alcohol mixture in the winter run, and proportion of isopropyl

alcohol was raised to 1 % in the summer run for additional quenching. Alcohol in the gas absorbed

19

damaging ultraviolet light at the signal amplification around the wire. With typical operating high

voltage of negative 2450 V – 2600 V on the cathodes and window wires, the drift velocity in each

chamber was approximately 50µm/ns.

Pulses from the sense wires were amplified and discriminated by preamplifier cards mounted

on the chamber. The discriminated signal was fed into LRS 3377 time to digital converters(TDCs)

operated in a common stop mode, that is, the incoming signal from the sense wire started the TDC

counting and all running TDCs were stopped by a common signal from a fast trigger(actually, Level

1 trigger explained later). The TDC time distribution is shown in Figure 2.4. The in-time window

was defined as 115 ns < t < 350 ns.

0 100 200 300 400 500time(ns)

In-time window

EarlyLate

1000

2000

3000

4000

5000

6000

7000

8000

Figure 2.4: TDC time distribution of drift chamber. The in-time window was defined as 115 ns

< t < 350 ns. The sharp edge at 350 ns was short drift time near the sense wire. The tail at 200

ns or less, which corresponded to a longer drift time, was due to hits very far from the sense wire

and non-uniformity of the electric field. The slope between 200 ns and 270 ns shows non-linearity

of the drift velocity far from the sense wire.

The sharp edge at 350 ns corresponds to a short drift time near the sense wire. The tail at 200

ns or less, which corresponded to a longer drift time, was due to hits very far from the sense wire

and non-uniformity of the electric field. The slope between 200 ns and 270 ns shows non-linearity

of the drift velocity far from the sense wire.

The drift times were converted to drift distances and then to the position of the track at the

chamber. For a hit pair of a proper single track, the sum of distances(SOD) of adjacent sense wires

as seen x1 + x2 in Figure 2.3, can be determined and it must be equal to the offset of adjacent

sense wires, 6.35 mm.

The SOD distribution is shown in Figure 2.5. There is a clear peak at 6.35 mm. The width

of the distribution, considered as a combined position resolution of two sense planes, was 150

µm assuming Gaussian distribution. It implies that the individual position resolution of the drift

20

SOD(mm)6.35 7.355.35

1000

2000

3000

4000

5000

6000

Figure 2.5: The sum of distance(SOD) distribution. The mean of SOD distribution corresponded

to a half of a drift cell, 6.35 mm.

chambers is 150µm/√

2 ' 100µm.

Analysis Magnet

The analysis magnet, a dipole magnet located between DC2 and DC3, provided a vertical magnetic

field. The strength of the magnet field was about 2000 gausses over an aperture of about 2 meters.

This gave a transverse momentum kick in horizontal direction of 205 MeV/c to charged particles.

Momentum Resolution

The momentum resolution of the KTeV spectrometer was measured to be a quadratic sum,

σp

p= 0.016%× p ⊕ 0.38%, (2.1)

where p is the momentum of a charged particle in GeV/c and σp is its deviation. The constant

term was due to the multiple scattering. The term linear in momentum is due to the finite position

resolutions of the drift chambers, whose contribution becomes larger for tracks with smaller bending

angle.

2.2.2 Trigger hodoscopes

Two planes of scintillator hodoscopes named V(upstream) and V’(downstream) were used at the

trigger level to select events with charged particles. The V hodoscope was located at Z = 183.90

m while V’ was located at Z = 183.95 m. Both banks had dimensions of 1.9 m × 1.9 m and 1.0 cm

thick. As shown in Figure 2.6, the scintillators were installed vertically and covered the fiducial

region. Beam holes were cut to reduce the beam interaction. The scintillation counters irregularly

21

V bank V’ bank

1.9 m 1.9 m

1.9

m1.9 m

Figure 2.6: The V and V’ trigger hodoscope planes. The V hodoscope located at Z = 183.90 m

while V’ was at Z = 183.95 m. Both banks had dimensions of 1.9 m × 1.9 m and 1.0 cm thick.

The scintillation counters had different sizes so that there was no overlapping gap between V and

V’ bank.

had different widths and lengths so that there was no overlapping gap between V and V’ banks.

This helped to reduce the overall inefficiency of V and V’ banks.

2.2.3 Electromagnetic Calorimeter

The KTeV CsI electromagnetic calorimeter primarily provided a precision energy measurement of

electromagnetic particles, e± and γ. In addition, the information of the relation between a particle

momentum and deposit energy in the calorimeter was used to identify electrons from pions.

The electromagnetic calorimeter was constructed of 3100 pure cesium iodide(CsI) crystals and

located at 186m from the target. The dimension of the calorimeter was 1.9 m × 1.9 m transverse

to the beams.

CsI Crystals

The calorimeter was stacked with 2232 small CsI crystals(2.5 × 2.5 × 50.0 cm) for the inner part

and 868 large CsI crystals(5.0 × 5.0 × 50.0 cm) for the outer part, as shown in Figure 2.7. The

depth of the crystals corresponded to 27 radiation lengths, to capture photon and electron showers

completely. The depth of crystals was also 1.4 nuclear interaction lengths so that most of hadrons

such as charged pions passed the calorimeter as minimum ionizing particles but some made a

hadronic shower in the calorimeter.

Each crystal was individually wrapped with 13µm-thick aluminized mylar and black mylar to

tune the response uniformity to the level of 5%.

The scintillation light from the CsI crystal was conducted to photomultiplier tubes(PMT)

22

0.5 m

1.9 m

1.9 m

Beam Direction

Figure 2.7: The KTeV electromagnetic calorimeter. The calorimeter had dimensions of 1.9 m(W)

× 1.9 m(H) × 0.5 m(D) and located at 158m from the target. The calorimeter was constructed from

3100 blocks of pure cesium iodide(CsI) crystals which were 2232 small CsI crystals(2.5× 2.5× 50.0

cm) for the inner part and 868 large CsI crystals(5.0× 5.0 × 50.0 cm) for the outer part.

23

instrumented on the downstream end of each crystal. PMTs were operated at -1200V high voltage

typically, resulting a gain of 5000. These PMTs showed a response linearity within the level of

0.5% at the typical operation. The opening window of the PMT was a UV transmitting glass to

accommodate the emission spectrum of the fast component of the scintillation light from the CsI

crystal.

The PMTs were mounted on the crystals with two layers of transparent RTV rubber filters,

and 1 mm-thick glass UV filter between them to remove slower components of the CsI scintillation

light.

Readout System

A digital PMT base(DPMT) system was used to read out the signals from the PMTs. One

DPMT card was attached just behind each CsI crystal. Since the DPMT digitized the signals

right after PMT, it enabled us noise-free data collection, without transmitting analog pulses for

long distance. The DPMT consisted of a high voltage divider for the PMT, a charge integrating

and encoding(QIE) circuit, an 8-bit flash ADC, and a “driver-buffer-clock”(DBC) circuit[14]. The

QIE and DBC devices were custom integrated circuits designed for this calorimeter.

The QIE was a hybrid digital/analog circuit integrating the charge from the PMT anode signal.

To achieve both wide sensitivity range and sufficient precision, the QIE, with the help of the flash

ADC, encoded the charge into an exponential form, i.e., 8 bits for the mantissa and 4 bits for the

exponent.

Each QIE had four identical circuits and used them in a round-robin manner. Since the circuit

was synchronized to the Tevatron RF and the operation was completed in one clock cycle, each

period of the charge integration was about 19 nsec, referred as a “slice.” This feature allowed

non-deadtime readout. The output of the QIE was the analog signal and sent to the flash ADC

for digitization.

Performance

Figure 2.8 shows a E/p distribution for electrons in Ke3 events, where E means a deposit energy

in the CsI calorimeter and p means the electron momentum measured with the spectrometer. The

intrinsic energy resolution of the calorimeter is shown in Figure 2.9 as a function of the momentum.

This can be roughly parameterized as a quadratic sum;

σE

E= 0.45% ⊕ 2%√

E(GeV). (2.2)

2.2.4 Photon Vetoes

In order to detect escaping photons and electrons away from the detector sensitive region, such as

the drift chambers and CsI calorimeter, ten photon veto counters were placed transverse to the

beam. They kept hermeticity for the decay products.

24

Energy/Momentum

Electrons from K→πeν

0

2000

4000

6000

8000

10000

x 103

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2E/P

Nevents = 1.92E+08σE/P = 0.75 %

Figure 2.8: E/p for electrons in Ke3 de-

cays, where E means an energy deposit in the

calorimeter, and p denotes a momentum mea-

sured by the spectrometer.

σ(Energy)/Energy vs. Momentum

Electrons from K→πeν

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0 10 20 30 40 50 60 70P (GeV)

σ(E

)/E

Figure 2.9: The calorimeter intrinsic energy

resolution measured in Ke3 events as a func-

tion of electron momentum.

25

Table 2.2: Positions and dimensions of the photon vetoes.

Elements Z position(m) Aperture(m) Outer Dimension(m)

RC6 132.596 0.84× 0.84 1.00 (radius)

RC7 138.598 0.84× 0.84 1.00 (radius)

RC8 146.598 1.18× 1.18 1.44 (radius)

RC9 152.600 1.18× 1.18 1.44 (radius)

RC10 158.599 1.18× 1.18 1.44 (radius)

SA2 165.116 1.540× 1.366 2.500× 2.500

SA3 173.985 1.692× 1.600 3.000× 2.400

SA4 180.018 1.754× 1.754 2.372× 2.372

CIA 185.191 1.842× 1.842 2.200× 2.200

CA 185.913 0.150× 0.150 0.180× 0.180

Ring counters(RC6-RC10) were located inside the vacuum pipe. These had a rounded outer

shape to fit the vacuum pipe and square apertures, and were azimuthally divided into 16 modules.

Each of them consisted of 24 scintillator-lead layers and approximately corresponded to 16 radiation

lengths.

There were three “Spectrometer Antis”(SA2-SA4) surrounding DC2-4, respectively. The “Ce-

sium Iodide Anti”(CIA) was also placed just upstream of the CsI calorimeter. These counters were

rectangular in both the inner aperture and the outer shape. The amount of the material of them

corresponded to 16 radiation lengths.

The Collar Anti(CA) was a veto counter which defined a fiducial area at the beam holes of the

CsI calorimeter. It consisted of the three layers of counters and 2.9 radiation lengths of tungsten,

and located in front of the calorimeter.

2.2.5 Hadron/Muon vetoes

At the downstream of the calorimeter, there were detector components to veto hadrons, muons

and particles through beam holes of the calorimeter. The dimensions and the Z positions of these

detectors are shown in Table 2.3.

Hadron Anti

A 15 cm thick lead wall was located Z = 188.5m, just downstream of the CsI calorimeter, to cut

off the small electromagnetic shower leakage from the calorimeter and to produce hadron showers.

A scintillator bank called “Hadron Anti”(HA) followed the lead wall, to veto events including

hadrons. There was a hole in the beam region of HA and the lead wall, to let neutral beams pass

through without interaction.

26

10

0cm

Figure 2.10: The schematic view of RC6, fac-

ing downstream. The beam passed through

the inner aperture. The perimeter of outer cir-

cle fit right with the vacuum pipe. The other

RCs were basically similar in a form but the

dimensions seen in Table 2.2.

23

7.2

cm

Figure 2.11: SA4 from upstream.

CA

Figure 2.12: The Collar Anti just in front of the calorimeter.

27

2.24m

2.24m

Figure 2.13: The HA hodoscope, which was

composed of the same size of 28 scintillation

counters. This followed the lead wall and de-

tects hadronic particles.

299c

m

378cm

Figure 2.14: The Mu2 counter.

Table 2.3: Positions and dimensions of the detectors(downstream of the calorimeter).

Elements Z position(m) Dimensions(m)

Lead wall 188.53(0.15m thick) 2.43× 2.43

HA 188.97 2.24× 2.24

Steal 1 189.09(1m thick) 2.4× 2.4

BA 191.09 0.60× 0.30

Steal 2 191.74(3m thick) 4.3× 3.4

MU2 194.83 3.93× 2.99

Steal 3 195.29(1m thick) 3.5× 3.6

MU3 196.36 3.00× 3.00

28

Muon Counter

Downstream of the HA, there was a stack of steel totaling roughly 5 m-thick followed by Muon

Veto Counter(MU2) which consisted of a scintillation hodoscope to reject events including muons.

The MU2 had large dimensions, 2.99 m × 3.78 m. The MU2 was used at the trigger level in this

analysis.

2.3 Trigger

This section briefly describes the trigger system in the KTeV experiment. Main purpose of the

trigger is to select signal events of interest and to reduce the event rate to an acceptable level for

the data acquisition system.

First, physics related triggers are explained. This trigger system had three levels; the first(Level

1) and second(Level 2) were formed by a hardware logic, and the third was software filtering(Level

3). The Level 1 trigger, which consisted of fast trigger sources, synchronized to the Tevatron RF(53

MHz) and provided a deadtimeless trigger. The Level 2 performed time consuming decisions such

as counting the number of hits in DCs, and the number of photons in the calorimeter. The

Level 3 trigger carried out complex event reconstructions including charged track trajectories,

energy clusters in the calorimeter, and simple vertex finding. The Level 3 filtering is explained in

Section 2.4, as a part of the data acquisition system. The trigger for the signal basically required

the existence of four charged tracks and allowed some additional particles such as photons.

In addition, calibration triggers were used to collect events necessary for the CsI calibration

and pedestal measurements.

2.3.1 Level 1 Trigger

In about 20 seconds of Tevatron spill, the primary proton beam and the secondary KL beam

arrived with a microstructure, called a bucket, 1 ∼ 2 nsec beam pulse once every 19 nsec. The

Level 1 trigger, based on fast trigger sources such as photomultiplier signals, was synchronized to

the microstructure and allowed deadtimeless triggering.

There were 80 Level 1 trigger sources in total which were made with NIM logic. Table 2.4

shows major Level 1 trigger sources used to take data for KL → π+π−e+e− and KL → π+π−π0D.

After resynchronizing to the Tevatron RF before triggering, the trigger sources were transfered to

twelve memory lookup units and decision was made simultaneously for 16 trigger outputs for each

RF clock cycle.

Next, we describe the Level 1 sources.

29

Table 2.4: The E799 four track trigger elements.

Trigger Element Trigger Level Description

GATE 1 Vetoes “fast neutrino pings”

3V TIGHT 1 ≥3 hits in both V and V’ banks

2DC12 MED 1 3 out of 4 DC1 and DC2 planes with ≥ 2 hits, 1 plane with ≥ 1 hit

ET THR1 1 ETOTAL ≥ 11GeV

MU2 1 Veto events with ≥ 1 hit (15 mV, 0.2 mip) in MU2

PHV 1 Veto events with ≥ 500MeV in PHV or events

with ≥ 400MeV in the SA

CA 1 Veto events with ≥ 14GeV in the Collar-Anti

34 HCY 2 require ≥ 3 hits in DC1Y, ≥ 3 hits in DC2Y

and ≥4 hits in DC3Y, ≥4 hits in DC4Y

HCC GE2 2 require ≥2 HCC clusters

YTF UDO 2 YTF trigger: Requires a good track in the upper half

and a good track in lower half or one good central track

3HC2X 2 ≥ 3 hits in DC2X Bananas

Neutrino Ping

When the KTeV experiment was running, Tevatron also provided the primary proton beam to

the other experiments. The “Neutrino Ping” signal was introduced to reject a fast, short-width

high intensity spill for a neutrino-scattering experiment. This is because if fast high intensity

beam happened to be delivered to the KTeV beam line, it could cause unacceptable high detector

activity.

Trigger hodoscope

The V and V’ were used to detect charged track hits. We required at least 3 hits in both the V

and V’ hodoscopes in this analysis.

DC-OR trigger

DC1 and DC2 were also used as a fast trigger source. Although the maximum drift time was about

200 ns, the track is always passing between two wires in two adjacent planes, so that one of the hits

should arrive within 100 ns after track passing. Therefore, we could utilize the logical OR of the

wires for the fast trigger source. The trigger source consisted of sets of 16 OR-ed sense wires(10.2

cm wide) and looked at the hit them in the X and Y views.

This trigger source called “DC-OR” was used to select events with charged particles passing

between DCs and V and V’ banks. The signal trigger required that three out of four planes had

30

at least 2 hits, and the rest had at least one hit.

Etotal

An analog sum of the signals from all PMTs in the CsI calorimeter was used as a Level 1 trigger

source. This approximately corresponded to total in-time energy deposit in the calorimeter. In

this study, the energy deposit was required to be over 11GeV nominal.

Muon Counters

The muon counters were also used to veto events. Since both the signal and normalization modes

do not include any muons, we vetoed events with any hits(0.2 mip) on the MU2 veto counter.

Photon Vetoes

Signals from the photon veto counters were also used to veto events with outgoing decay products

from the fiducial volume. We vetoed if photons or electrons hit the photon vetoes and had an

energy deposit of 500MeV or larger in the ring counters(RC6-10), or 400MeV or larger in the

spectrometer anti(SA2-4). For the collar anti(CA), events were vetoed if particles hit CA and

there was an energy deposit of over 14GeV.

2.3.2 Level 2 Trigger

The Level 2 trigger system is briefly explained here. Details can be found in Ref. [15] if one needs

more explanation.

The Level 2 trigger consisted of a slower trigger logic. This allowed us to do a simple event

reconstruction for further background and accidental event reduction. When an event was accepted

by Level 1, the Level 2 trigger decided if the event should be accepted to be sent to the DAQ/Level

3 system, or not.

Drift Chamber Activity(Kumquat)

This trigger required hits which were identified as proper track hits in Y view of drift chambers.

In this analysis, the drift chamber hit in a 205 ns gate was required in Y view for the trigger level;

at least 3 hits each in DC1Y and DC2Y, and at least 4 hits each in DC3Y and DC4Y.

Hardware Cluster Counter

The hardware cluster counter [16] counted the number of in-time clusters in the CsI calorimeter.

The hardware cluster was defined as a cluster consisting of crystals which had the energy deposit

of 1GeV or more with a shared perimeter. Therefore, the particle associated with the hardware

cluster was identified to be a electron or photon. This counting was achieved with a hard-wired

31

energy cluster search algorithm and PMT outputs. We required at least two hardware clusters in

the event.

YTF trigger

The function of Y Track Finder(YTF) [17] trigger made a simple Y track reconstruction by looking

at hit-patterns in the drift chamber. For the momentum balance, we required the existence of

reconstructed tracks in both upper and lower halves of chambers.

Sum of Distance Correlation(Banana)

This trigger equipment was used to require events with in-time hits in the drift chambers. The

hits were assumed to be in-time if the sum of distances was in an allowed region. The time of each

hit was measured with 625 MHz TDCs.

At least three proper hits in DC2-X were required in this analysis.

2.3.3 Trigger Rate

During the winter run, the beam intensity was about 4 × 1012 protons per spill, where the beam

duration was about 19 sec. For four track trigger containing KL → π+π−e+e− and KL → π+π−π0D,

the trigger rate was about 25 kHz for the Level 1 trigger and about 2 kHz for the Level 2 trigger(70

kHz for Level 1 and 7 kHz for Level 2 of total trigger rate). The DAQ live time was typically 60–

70%, depending on the beam rate and instantaneous beam intensity.

2.3.4 Calibration Trigger

The calibration trigger involved the following: laser calibration of the CsI calorimeter; pedestal

measurements in all the detectors; uniformity measurements of the CsI by using cosmic muons.

They were generated from local logic circuits.

The trigger of laser calibration was formed by a flash of laser/dye system(5 Hz on- and off-spill).

The pedestal was collected by opening ADC gates intentionally both on-spill and off-spill at

the rate of 3 Hz.

There were additional muon telescopes located at the top and bottom of the CsI blockhouse

for triggering and positioning the cosmic muon event. The rate of this trigger was about 30 Hz,

and it was taken during off-spill only. These events were used to measure the CsI uniformity in

response along the Z direction.

2.4 Data Acquisition System

After performing event selection at the L1/L2 triggers, data were sent to KTeV data acquisition

system(DAQ). In the KTeV experiment, the event rate after the Level 2 trigger was approximately

32

11 KHz and the average event size was 8 KBytes nominal, equal to about 2 GBytes per spill(∼11KHz×8KBytes×20sec/spill). The system was designed to handle twice this level of bandwidth.

Here, we glance over the DAQ and Level 3 filtering. More details are found in Ref. [18, 19].

2.4.1 Frontend and DAQ

Memory Memory

Memory

Memory Memory MemoryMemory

SGIChallenge

Challenge

SGIChallenge

SGI

SGI

4 Planes

6 Streams

Challenge

Memory

1st Event sub-sub-event 2event 1

sub-event 3 event 4 event 5

sub-event 6

sub-

2nd Event

Memory MemoryMemoryMemory Memory

Memory Memory

Memory

Memory MemoryVME

VME

sub-

VME

VME

Memory MemoryMemory MemoryMemory Memory

RS485 lines

Figure 2.15: Data acquisition system. Three planes were used for Level 3 trigger and the other

was for online monitoring of the detector performance and online calibration.

In Figure 2.15, the basic structure of the DAQ is drawn. An event, which consisted of six

subevents at the frontend, was asynchronously read out through a fast data bus. The data was then

sent to buffer memories(streams) through six RS485 lines, whose bandwidth was 40Mbytes/sec/line

each. The buffer memory had a capacity of 4GB, large enough to store all the events collected

over a spill, approximately 2GB.

Each stream formed “plane”, which was logically connected to a Unix workstation(SGI Chal-

lenge). The Unix work stations performed the event reconstruction for the filtering and monitoring:

Three of four planes were used for the Level 3 filtering, and the other was used for monitoring of

the detector response and online calibration.

2.4.2 Level 3 filtering

The level 3 filtering provided a simple event reconstruction with a software filtering on SGI Chal-

lenges. This allowed us to reduce the trigger rate based on the information from sets of detector

elements such as charged track and vertex reconstruction, particle identification with calorimeter

E/p, and muon counter hits.

33

The Level 3 filter for our study required three reconstructed tracks shared a vertex which

located from 90.0 to 158.0 meters from the target.

2.5 Physics Run

This section describes the data collected in the KTeV-E799II runs, which was used for physics

analysis of KL → π+π−e+e− and KL → π+π−π0D.

The data used in this measurement consisted of two runs of KTeV-E799II experiment, winter

run and summer run. The winter run was performed between February 1 and March 23, 1997,

run 8245 through 8910. The summer run was between July 24 and September 3, 1997, run 10463

through 10970.

There were some differences between the winter and the summer runs, as shown in Table 2.5.

The main difference between them was the proton beam intensity and neutral beam size. The

proton beam intensity was lowered during the summer run due to the sharing of the proton beam

in the Tevatron with other experiments. To keep appropriate kaon beam intensity, the larger

neutral beams were used in the summer run to compensate for the decreased intensity of the

primary proton beam.

Table 2.5: The KTeV-E799II run conditions.Winter Run Summer Run

Run Number 8245-8910 10463-10970

Proton Intensity(spill) 5.0 × 1012 3.5 × 1012

Beam Size at the CsI 10 cm×10 cm 12 cm×12 cm

4 Track L1 Trigger Rates 20 KHz 23 KHz

4 Track L2 Trigger Rates 1.5 KHz 2.3 KHz

The collected data were written to about 500 Digital Linear Tapes(DLT) at the experiment.

Each DLT stored about 10GB of data. Since they included all physics triggers, we split off only 4

track tagged events into 38 DLTs for the winter and 44 DLTs for the summer runs. Furthermore,

the data of 82 4 track triggered tapes were processed in a step. This separated and filtered the

4 track triggered events into datasets which were used for a different analysis. The filtering was

performed with offline event reconstruction code. This stage reduced the total data size to 17

DLTs. In our analysis, we analyzed events based on this data set.

34

Chapter 3

Monte Carlo Simulation

In each phase of this analysis, such as the acceptance calculation in both the signal and the

normalization mode, the background estimation, and the systematic study, we strongly relied on

the Monte Carlo simulation(MC) to understand the detector performance and particle decays’

kinematics. In the Monte Carlo simulation, we simulated the beam production, particle decays,

detector response, signal digitization and the triggering. Since output data format from the Monte

Carlo simulation was the same as that from the real KTeV detector, we could analyze real data

and the Monte Carlo events with the same event reconstruction algorithm and to make detailed,

synthetic study on both of them.

The Monte Carlo simulation was processed in five stages: Kaon beam production, decay gener-

ation, tracing in the detector, signal digitization, and trigger. We will glance them in this chapter.

3.1 Kaon production

The neutral kaons were produced in the target, based on the measured beam position and width.

In the generation, we used a parameterization by Malensek [7] for the momentum spectrum of

K+ and K− produced by protons incident onto a beryllium target. In the parameterization, the

number of kaons with momentum p into a solid angle dΩ at a polar angle θ was proportional to:

d2N

dpdΩ=

B

400· x · (1 − x)A(1 + 5e−Dx)

(1 + p2t /M

2)4, (3.1)

where p is the produced particle momentum and EB is the beam energy, x = p/EB, and pt repre-

sents the transverse momentum of the produced particle relative to the incident beam direction.

The B, A, D, and M2 were determined from the experimental data obtained by 400 GeV/c proton

beam, and their values are shown in Table 3.1.

The production spectrum for the neutral kaon can be derived from that for charged kaons. Let

us define σu, σd, and σs, as the production probabilities of uu, dd and ss quark pairs, respectively,

and let us assume σu = σd. The production of a K+ needs the creation of a ss. In this case,

35

Table 3.1: The parameters referred from Malensek [7]

K+ K−

A 2.924 6.107

B 14.15 12.33

M2 1.164 1.098

D 19.89 17.78

since one of the two valence u quarks in the proton beam couples to the s quark, the production

probability of K+ is proportional to 2σs. We can also consider another case, where a u quark is

taken from the sea whose probability is proportional to σsσu. Similarly,

σ(K+) ∼ 2σs + σsσu (3.2)

σ(K−) ∼ σsσu (3.3)

σ(K0) ∼ σs + σsσu =σ(K+) + σ(K−)

2(3.4)

σ(K0) ∼ σsσu = σ(K−). (3.5)

This provides an approximation of the K0 and K0 production spectra. The spectrum was

tuned further to match kaon momentum measured by KL,S → π+π− events in the data. The

correction factor, ξ(p) was

ξ(p) = 1 + 10.655x− 55.337x2 + 60.033x3, (3.6)

where x = p(GeV/c)/1000. This was multiplied as a factor of the generation probability:

d2N

dpdΩ(K0) =

12[d2N

dpdΩ(K+) +

d2N

dpdΩ(K−)] × ξ(p),

d2N

dpdΩ(K0) =

d2N

dpdΩ(K−) × ξ(p). (3.7)

Both K0 and K0 were produced according to Equation 3.7 in our Monte Carlo simulation.

Figure 3.1 shows the momentum spectrum of generated kaons.

After the production of a K0 or K0, it was propagated to a decay point in the vacuum volume

of the detector by taking into account the full amplitude evolution. The decay position was chosen

based on its lifetime and momentum. The distribution of the decay position of KLs is shown in

Figure 3.2.

3.2 The Decays

After the decay position was determined, the Monte Carlo simulation generated decay products of

KL. Here, we briefly explain the decay processes used in the Monte Carlo simulation.

36

Momentum of Generated Kaons ΓεΧ/η

0

200

400

600

800

1000

1200

x 103

20 40 60 80 100 120 140 160 180 200 220

Figure 3.1: Momentum distribution of gener-

ated kaons at the target.

Z Position of Generated KL Decays µετερσ

0

1000

2000

3000

4000

5000

6000

7000

8000

x 102

90 100 110 120 130 140 150 160

Figure 3.2: KL decay Z-distribution of Monte

Carlo generated kaons.

3.2.1 KL → π+π−e+e−

The matrix element to generate the decay KL → π+π−e+e− completely relied on the equation in

Ref. [22, 23], as described in Section 1.3. The direct CP violating effect, however, was eliminated

since the effect on this analysis was expected to be negligible. The coefficients in the matrix element

were obtained from PDG [9] and theoretical estimation from Ref. [22, 23].

In addition, we also modified the form factor in direct photon emission component gM1, as in

the recent study [29, 37, 38],

gM1 −→ gM1 · FF =

a1

(M2ρ − M2

K) + 2MKEγ∗ + a2, (3.8)

where Eγ∗ is the energy of the parent photon of the electron-positron pair in the kaon center-of-

mass system, and Mρ is the mass of the neutral ρ meson. We independently determined a1/a2

from the kinematics of KL → π+π−e+e− in Chapter 6.

3.2.2 KL → π+π−π0

The KL → π+π−π0 decay was simulated with a pure phase space distribution. Since the π0 flies

too short(cτ = 25.1nm) to trace with the KTeV detector, it immediately decayed to γγ or e+e−γ

in the Monte Carlo simulation.

In case of the Dalitz decay, π0 → e+e−γ, the generation was based on the formulation by Kroll

and Wada [43], including O(α2EM) radiative corrections by Mikaelian and Smith [44, 45], and inner

bremsstrahlung corrections(π0 → e+e−γγ) and virtual-photon corrections.

37

3.2.3 Particle Tracing

After a kaon decay was generated and π0 decayed into γγ or e+e−γ immediately, decay products

were boosted to the lab frame and traced through the detector. Tracing was terminated if the

particle decayed, interacted with materials, or went out from the detector. All decay products

except π±’s were stable so they were propagated without decaying. For charged pions, the proper

lifetime was assumed and some of them decayed into µν in flight.

3.3 Detector

Charged particles passing through the detector material could be scattered by the Coulomb multiple

scattering according to a parameterization of the Moliere theory [46]. This also accounted for a

scattering angle distribution of a non-Gaussian tail caused by the single scattering effect. For

electrons, in addition to the scattering, radiative bremsstrahlung was allowed according to the

Bethe-Heitler cross-section. Radiated photons could be converted in material with probability

1−e−79 (X/X0), where X/X0 is the amount of the material in radiation lengths. The energy spectrum

of the converted electron-positron pair was defined by Bethe-Heitler formula, and routines in EGS4

electromagnetic shower library [47] was imported for calculating the opening angle of electron and

positron pair.

Generated particles in these interactions were also traced as well as daughter particles of the

decay.

When a particle escaped from the fiducial volume of the detector, defined by the outer edge of

detector elements, it was not traced further, to save the processing time.

3.3.1 Photon Veto

Charged particles, except electrons, were treated as minimum ionizing particles, when they pass

through photon veto counters, and an energy deposit of the incident particle was smeared with

Gaussian distribution. The parameter of the energy deposit distribution was determined with cal-

ibration constants acquired from calibration runs. Minimum ionizing particles were also scattered

here.

For electrons and photons, the particle was stopped when it entered a photon veto, and all of

its energy was deposited in the detector after Gaussian smearing.

A trigger was vetoed if the total deposit energy of one of the counters, exceeded its threshold.

The energy deposit was digitized to ADC counts based on a gain determined with calibration runs.

3.3.2 Drift Chamber

When charged particles pass through the drift chambers, the incident position was stored and the

drift distance, which is the distance between the particle trajectory and the closest sense wires in

the chamber plane, was calculated. A drift time was translated from the drift distance with the

38

calibration data, and smeared to the chamber resolution of 100µm. The drift time was converted

into TDC counts with certain offsets. Only the first hit on a wire in multiple hits was simulated.

The δ-ray effect which causes low-SOD pairs were simulated in the chamber gas. This was

treated as multiple hits on the same cell.

3.3.3 CsI calorimeter

The CsI calorimeter was simulated for all particles entering to the front face of the calorimeter.

For photons, electrons, and pions, the simulation determined the incident position and generated

a shower. Muons were treated as minimum ionizing particles. After calculating deposit energy for

each crystal in an event, the response of scintillation light was simulated, and its digitization was

done for each crystal. The trigger component Etotal and HCC were instrumented in the Monte

Carlo simulation.

Electromagnetic Shower

When electrons and photons were propagated to the face of the CsI crystal, its extrapolated

transverse position at the depth of the shower mean was determined. The depth was calculated

with:

Ze(m) = 0.11 + 0.018× ln E(GeV) for electrons, (3.9)

Zγ(m) = 0.12 + 0.018× ln E(GeV) for photons, (3.10)

where E is the incident energy of electron(photon). The CsI calorimeter has 27 radiation lengths

so that electron or photon showers were well-contained in the calorimeter. Therefore, tracing was

terminated at the calorimeter for those particles.

With the information of the incident position and the incident energy, the energy deposit in the

crystals was determined by samples(shower library) premade by GEANT simulation. Each single

event in the shower library was composed of energy maps on 13×13 small(2.5cm×2.5cm) crystals.

The large crystal(5.0cm×5.0cm) was simulated as four small crystals.

The events in the library were classified into bins according to the incident energy, 2, 4, 8, 16,

32, and 64 GeV. In addition, the incident position of the face of the CsI crystal were binned into

50 × 50 bins, to simulate the position resolution at sub-millimeters with reconstructing an energy

cluster.

The energy deposit were segmented longitudinally into 25 slices to simulate the light output

from a crystal with nonuniform behavior. The uniformity was calibrated for each crystal with

radioactive source in advance, and cosmic muons during runtime.

Hadronic Shower

The hadronic shower in the CsI calorimeter was also simulated in the Monte Carlo. This is

because the shower shape and the energy deposit in the CsI calorimeter are used to distinguish

39

E/p

1

10

10 2

10 3

10 4

0 0.2 0.4 0.6 0.8 1

Figure 3.3: E/p distribution of pion shower samples(actual data from Ke3 and Monte Carlo simu-

lation from hadronic shower library).

electrons from pions in the analyses, and satellite clusters of the hadronic shower can mimic photon

clusters(Figure 3.4).

In order to avoid time consuming event by event shower simulation, a large sample of hadronic

showers was first produced and stored in a library to be used later in the Monte Carlo event

generation. The shower library for the KTeV detector simulation was generated using GEANT

3.21 with FLUKA for the hadronic interactions. In the generation, a pion was used as an incident

particle and cut off energy in the CsI was 0.001 GeV for hadronic and electromagnetic interactions.

Pions were injected uniformly on the surface, and its incident position was binned into 0.25 cm

by 0.25 cm square regions. The incident energy is also divided into 12 bins from 2 GeV to 64 GeV

in the library. Each event in the library had energy deposit in 13 by 13 array of 2.50 cm crystals,

and in the hadron anti counters. The crystals were segmented along beam axis into 25 slices to

obtain the energy cluster information in depth. The energy deposit in each slice was stored in the

library.

When generating events in the Monte Carlo simulation, if a pion hits the CsI, a shower sample

from an appropriate incident energy and position bin is selected randomly from the shower library.

The energy deposits in each slice in depth are scaled to the energy of the incident particle and

convoluted with measured CsI scintillation response to calculate the total charge from PMTs.

As shown in Figure 3.3, E/p distribution of the Monte Carlo simulation with the shower library

is consistent with that of actual hadron showers from Ke3 decay.

40

Electromagnetic Showers

Hadronic Showers

> 10.00 GeV

> 1.00 GeV

> 0.10 GeV

> 0.01 GeV

Figure 3.4: Electromagnetic and hadronic shower shape comparison in the Monte Carlo. The

electromagnetic shower forms round while the hadronic shower has an irregular shape.

41

For muons, the deposit energy in the CsI crystals was calculated as a minimum ionizing particle.

Digitization

After deciding the energy deposit in each crystal, the Monte Carlo simulation proceeded to the

digitization. The time structure of the PMT pulse and the digitization at the DPMT were sim-

ulated. These procedures included appropriate smearing due to simulate photostatistics effects.

The energy was converted into charge assigned to each DPMT time slice with constants for each

crystal. Constants were calculated from electrons in real Ke3 events.

3.4 Accidentals

Actual detectors have accidental activities, caused by the electronic noise in the detector and

contaminations in the neutral beam, mostly neutrons scattering or interacting with materials of

the detector. This section describes how the accidental activities were simulated.

3.4.1 Accidental Effect

The accidental activity faked hits on the detectors. For KL → π+π−e+e− study, there were two

major effects on the analysis. First, a signal acceptance decreased with accidental hits. This is

because accidental hits on the photon veto and muon veto counters could fire the vetoes. This effect

was also found in the event selection in the analysis. Second, the accidental activities contributed to

the misreconstruction of the events. This effect mainly increased the background level. Therefore,

accidental activities should be simulated in the Monte Carlo simulation.

3.4.2 Accidental Overlay

While the Monte Carlo simulation cannot reproduce such activities precisely, these accidental

activities are independent of the kaon decay in nature. Therefore, the Monte Carlo simulated

the accidental activity by overlaying accidental hits collected from the actual run. The accidental

trigger served for this purpose. This trigger was a random trigger, whose rate was proportional to

the instantaneous proton rate. The accidental trigger was taken during the experiment.

After the event generation in usual manner on the Monte Carlo, the accidental overlay was

performed at the digitization: ADC counts from accidental hits were simply added to the Monte

Carlo simulation. The latch hits of detectors were OR-ed, and the earliest TDC hit was kept.

Trigger requirements were then tested for each component again.

42

Chapter 4

Event Reconstruction

In KL → π+π−e+e− analysis, it is important to find charged tracks, the decay vertex of KL,

and to identify the particles. Event reconstruction, which gives these information, was done by a

combination of track finding with the spectrometer and the CsI calorimeter. In this chapter, we

describe how events were reconstructed in the analysis.

The details in the reconstruction procedure are described in the following sections. The event

reconstruction sequence is:

• Finding candidates of tracks from the drift chamber hits.

• Energy cluster finding on the CsI calorimeter.

• Track reconstruction with the information of the track candidates and the energy clusters.

• Vertex finding.

• Particle identification with the track momentum and cluster energy.

4.1 Track Candidate Finding

Track candidates were reconstructed with chamber hits in this phase [48]. Since tracks were bent

at the analysis magnet in X(horizontal) view but not in Y(vertical) view, tracks were reconstructed

in X and Y view separately.

4.1.1 Hit Pairing

First, hit pairs in each plane of the drift chambers were searched for in X and Y planes inde-

pendently. A hit pair was defined as a hit on a sense wire and a neighboring hit in the adjacent

plane.

The hit pairs were classified for their quality. Figure 4.1 shows possible cases. The “good SOD”

pair was defined as the SOD close to a half the drift cell size, |SOD − 6.35mm| < 1.0mm. The

43

“low SOD” pair having SOD less than 6.35 mm was also accepted since a hit pair with low SOD

generally occurred when two or more tracks happened to pass through a cell. A “high SOD” pair

was discarded in this analysis. Because the drift chambers had a small intrinsic inefficiency, it is

also necessary to identify an “isolated hit” without a neighboring hit, to use them to form a track

candidate.

In order to evaluate the quality of the reconstructed track later, a number called “pair value”

was assigned to the good-SOD, low-SOD and isolated hit pair as 4, 2, and 1, respectively.

Good SOD Low SOD Isolated

Figure 4.1: Schematic view of the classification of hit pairs. Dots indicate the sense wires. The

solid arrow means the trajectory of a charged particle, and a hair line between a trajectory and a

sense wire shows drift distance reconstructed from TDC counts. A good SOD pair could define a

trajectory while an isolated hit had a two-way ambiguity. The evaluation value called “pair value”

was assigned to these hit pairs as 4, 2, and 1, respectively.

4.1.2 Finding Y track candidates

After hit pairs were found, Y track candidates were searched for by picking a hit pair in DC1 and

DC4 each, and interpolating them with a straight line. If hit pairs in both DC2 and DC3 lied within

1 cm from this line, those hit pairs were fitted for a straight line and the χ2 was calculated. In

addition, the sum of the pair values was required to be at least 11, to select good track candidates.

4.1.3 Finding X track candidates

The X track had a kink at the magnet placed in the middle of the spectrometer. For this reason, X

tracks were reconstructed in upstream and downstream segments of the spectrometer, separately.

The pair value was also required as greater than 3 in both upstream and downstream X track

reconstruction.

The upstream and downstream track segments were then extrapolated to the center of the

analysis magnet at Z = 170 m. If the distance between the two track segments was less than 0.6

44

cm, and the sum of the pair values of both segments was 11 or more, they formed an X track

candidate.

4.2 Cluster Finding

In the CsI calorimeter, the cluster energy was defined as a sum of energy deposits in a 7×7 array of

small crystals or 3×3 array of large crystals centered on a crystal with the local maximum deposit

energy, which is called “seed crystal.” Energy clusters in the CsI calorimeter were classified into

two different categories; “hardware cluster”, and “software cluster.” The only difference was the

determination of the seed crystal in the cluster. In this analysis, the hardware cluster was formed

around a seed crystal with the HCC bit on. The software cluster had a seed crystal with a deposit

energy of 100 MeV or more and without the HCC bit, and the cluster energy was 250 MeV or

more.

Therefore, electromagnetic particles such as a photon and electron must be associated with

the hardware clusters while minimum ionizing particles such as a pion and muon, which typically

deposited 350MeV, mostly produced software clusters.

After clustering, a sequence of corrections were made to the cluster energies. This included

cluster overlap correction, boundary correction between small and large crystal region, missing

crystal correction around the beam holes and the outer perimeter.

In order to determine the X(Y) position of the cluster, the sum of energies in the central col-

umn(row) and the adjacent columns(rows) were calculated. The energy ratio between the adjacent

column(row) with larger energy and the central column(row) was used to get the cluster position

within the central block, by looking up a table generated with GEANT. Typically, the position

resolution for electromagnetic clusters was 1 mm.

4.3 Track-Cluster Matching and Vertex Finding

After cluster finding, the correct combination of track candidates in X and Y views was chosen by

matching the cluster to a track position on the CsI calorimeter. The distance between the track

X-Y position at the shower maximum and the cluster was required to be less than 7.0 cm. The

combination of the X and Y view tracks with the smallest track-cluster separation was considered

as the correct matching.

The momentum of a reconstructed track was measured by using the track bending angle in the

X view and the pt kick of the analysis magnet.

We proceeded to the vertex finding if four well-defined tracks were found. To obtain the vertex

position, we looked at the Z position of the intersection of X-tracks and Y-tracks, and then we

evaluated the vertex quality with χ2 calculation.

Once the vertex position was confirmed to be inside the decay volume, various corrections were

made on the hit positions in each chamber, in order to measure kinematic variables precisely.

45

These included the fringe field of the analysis magnet, chamber rotation, propagation time devi-

ation by the hit position. The vertex position was finally determined from the fit with charged

tracks weighted according to their multiple scattering angles and the errors in the hit position

measurement in each chamber. For some events with multiple vertex position candidates, a proper

vertex was chosen by looking at the fit quality of the vertex and tracks. In this analysis, the

four-track vertex resolution was about 0.5 m in Z.

4.4 Particle Identification

Particle discrimination of charged pions and electrons are critical in the asymmetry study of the

decay KL → π+π−e+e−. The e/π separation was done by using the information from the clus-

ter energy E and the track momentum p. Electrons deposited most of their energy in the CsI

calorimeter while pions deposited a part of their energy. This is because the CsI calorimeter had

27 radiation lengths, enough to contain all of the electron energy, but only 1.4 interaction lengths

for pions.

Figure 4.2 shows a E/p distribution for electrons and pions of Ke3 decays, where particles

were identified by transition radiation detectors(TRDs). Electrons have a sharp peak at unity,

while pions have a peak around zero with a gentle tail. A small peak at zero for electron was

due to pions and muons misidentified by TRDs. In this analysis, we required the E/p to be unity

within ±10% for electrons, and less than 0.9 for pions. In the electron identification, 99.6 % of

electrons were accepted while 99.3 % of pions were rejected after this cut. This means the particle

misidentification rate was only 0.7 % level with the E/p particle identification method. For further

improvement of the particle identification, the KTeV detector was instrumented with transition

radiation detectors(TRDs), although we did not use it for our analysis since we did not have any

significant background originated from the particle misidentification.

Muons were vetoed by MU2 counter at the triggering level. No further suppression of muons

were necessary because there was no significant background due to muons.

46

E/p Comparison

1

10

10 2

10 3

10 4

10 5

0 0.2 0.4 0.6 0.8 1 1.2

Figure 4.2: E/p comparison between electrons and pions of Ke3 decays. Particles were identified

by transition radiation detectors(TRDs). Electrons have a sharp peak at unity while pions have a

peak around zero with a gentle tail. A small peak at zero for electron was due to pions and muons

misidentified by TRDs.

47

Chapter 5

Event Selection

This chapter describes the event selection to suppress various types of backgrounds.

First, we summarize potential background sources remaining in the dataset after the triggering

phase. Next, an event selection scheme to suppress backgrounds while keeping signal events is

described. Finally, to estimate the absolute background level from each source, we explain the

normalization procedure, which includes KL flux measurement by using the decay KL → π+π−π0D.

In this chapter, to estimate the background level and signal acceptance, we used the Monte Carlo

simulation described in Chapter 3.

5.1 Background

Here, let us classify background sources potentially contaminating the signal region. This helps

us to decide a policy to reduce each background effectively. Generally speaking, backgrounds are

complex compositions caused by a kinematical similarity, a lack of information from the detector,

a finite resolution and smearing at the detection, accidental activity, and misreconstruction in the

analysis. They should be suppressed with appropriate constraints.

The main background sources are classified into three groups:

1. Dalitz background. Charged tracks are reconstructed while one or more photons are missed.

KL → π+π−π0D

KL → π+π−π0DD(π0 → e+e−e+e−)

Ξ → Λ(→ pπ−)π0D

2. Conversion background. A photon from decay products converts to an electron-positron pair

at material.

KL → π+π−π0

KL → π+π−γ

3. Double decay background. Two or more simultaneous decays in the detector happen to be

48

consistent with the signal.

Simultaneous decays of KL → π+e−ν and KL → π−e+ν

These background sources are classified with the distinctive features of decay kinematics and

topology from those of the signal. Therefore, different treatments are required for them. Here, let

us consider how to suppress those background decays.

• KL → π+π−π0D

This background was most serious in this study since it has a large branching ratio(1.5×10−3)

compared with the signal(3×10−7). Especially, the event topology is exactly the same as the

signal if the photon is missed or escaped from the detector or overlapped on the other cluster.

The difference from the signal shows up in a finite transverse momentum squared of the four

charged tracks, the invariant mass reconstructed with charged tracks, and an existence of a

photon-like energy deposit in the CsI calorimeter. For further suppression, we introduced a

constraint on a kinematical variable, named “Pp0kine”, described later.

• KL → π+π−π0DD(π0 → e+e−e+e−)

Although this decay mode has small branching ratio(4× 10−6), this mode is also a potential

background, since two slow electrons in the products of π0 → e+e−e+e− could easily be

kicked out from the fiducial volume by the analysis magnet. The existence of two missing

particles means that the invariant mass cut around kaon mass can limit this mode.

• KL → π+π−π0

This decay will give the same decay topology of KL → π+π−π0D if one of the photons from

π0 decay converted at material such as the vacuum window. This kind of background is

named as a “conversion background.” Fortunately, material upstream of the DC1 was kept

very little, and the opening angle of the converted electron-positron pair is small enough

so that we can identify and reject those effectively. Besides, this background has similar

characteristics of the decay KL → π+π−π0D.

• KL → π+π−γ

Another conversion background is from the decay KL → π+π−γ. This is potentially a

background since the track topology is identical to the signal when the photon converts. As

well as the other conversion background, this also has a small opening angle between the

electron-positron pair, so the same cuts to reduce the conversion background will work well.

• Ξ → Λ(→ pπ−)π0D

Since the KTeV detector could not distinguish π± from a proton, this decay could be a

background source. This decay mode was characterized with a large momentum of the Ξ and

an offset vertex of Λ → pπ− from Ξ → Λπ0D. A vertex quality cut and a constraint on the

KL momentum can reject this background.

49

• K → π±e∓ν simultaneous double decays

Simultaneous double decays of K → π±e∓ν gives the same charged track topology of the

signal. Since their two vertices have no correlation, we can reject these events by looking at

the vertex quality, as well as other kinematical constraints.

5.2 Basic Constraints

Before proceeding to the specific background rejection, we will apply basic constraints to the

dataset, in order to verify the trigger constraint, and to define the fiducial volume of the KTeV

detector. This reduces the uncertainty from the KTeV detector and the backgrounds, caused by

the discrepancy between the detector simulation and the real detector.

5.2.1 Fiducial Volume Cut

In order to certify events from proper KL decays, we required fiducial volume constraints for all

events. The constraints should be made not only to define the decay volume and detector sensitive

materials, but also to reject misreconstructed events.

• X-Y Vertex Constraint

The X-Y position of the reconstructed vertex projected onto the calorimeter from the target

should be within the squares of 14 cm × 14 cm centered at the beam-center, regardless of

the beam size. This reduced events caused by interactions at materials, such as photon veto

counters.

• Z Vertex Constraint

The vertex must be between Z = 95 m and 154 m. At Z = 160 m, the vacuum window

was placed and it could cause backgrounds from photon conversion and beam interaction.

Around Z = 90 m, the sweeper magnet and the defining collimeter were located, so this cut

reduced events with products originating in them.

• Track hit position on the CsI calorimeter

When photons and charged tracks hit near the perimeter of the CsI calorimeter, the deposit

energy of a particle would leak outside of the calorimeter. This caused an inaccurate energy

cluster reconstruction. This means that the energy deposit calculation, track-cluster associ-

ation, and, E/p might be unreliable around the perimeter. For this reason, the collar anti

was served to veto hits around the beam holes. We made a verification that tracks did not

point to the collar anti. In addition, we vetoed events with hits within 2 cm from the outer

edge of the calorimeter.

50

5.2.2 Trigger Verification

Constraints were made on the data sample to ensure that events should satisfy the Level 1 and

Level 2 trigger. This rejects the event which passed the trigger with extra accidental activities in

the detector, even if the event could not have by itself. These “volunteer” events could not be

simulated accurately in the Monte Carlo, so they should be removed from the data sample. The

trigger verification was applied to the Monte Carlo events, also.

The trigger requirements were shown in Section 2.3. The required trigger verification was:

• There were at least three in-time hits in both V and V’ banks which were associated with

tracks. The in-time hits in the counter were identified with TDC timing.

• The minimum energy of the hardware cluster was required to be at least 2GeV. This sup-

pressed the effect from the fluctuation of the HCC thresholds.

• Two electrons should have a total energy deposit in the CsI calorimeter of more than 11GeV.

This verified the Etotal trigger processor.

• The deposit energies of the photon vetoes were required to be below their online thresholds.

• The muon veto counters were required not to have any in-time hits. The in-time hit was

identified with TDC timing. Also, the threshold of counters was set the same as the online

trigger.

5.2.3 Consistency Check

As further basic cuts, we checked that the events were consistent with the KL → π+π−e+e− signal,

as follows:

• The number of charged tracks was required to be exactly four, and all the tracks should share

the same vertex. This requirement rejected any background with accidental hits or tracks.

• Four charged tracks should point to the CsI calorimeter. This gave us well-identified tracks.

• Events with extra hardware clusters, which had a deposit energy of 2GeV or more, were

rejected. Such extra clusters were identified as photons.

• No cluster sharing was allowed. Since the particle identification was made with a cluster

energy and a track momentum, a shared cluster gave wrong information of the deposit

energy. The distance between two tracks at the shower maximum in the calorimeter should

be 8 cm or more, for the same reason.

• Particles should be identified as π+π−e+e−, including the charges. A half of Ke3 double

decay background would be rejected with this cut.

• Since the reconstructed KL momentum of signal was mostly populated over 40 GeV/c, we

reject events with a momentum below 40 GeV/c.

51

5.3 Event Selection

After applying the basic constraints, we turn to background suppression against the individual

background sources. Some backgrounds have been already suppressed with the basic cuts, but still

background events were dominant in the data sample. The background feature was simulated well

by the Monte Carlo in this phase, so that we compared the Monte Carlo to the data sample, in

order to understand the kinematic feature of backgrounds.

An accumulation plot in Figure 5.1 shows the background contribution from each source overlaid

onto the actual data. Note the plot was made of events with only p2t < 0.0001GeV2/c2, to emphasize

the signal peak and clarify the background contribution. Most of the events in the lower tail,

Mππee < 0.48GeV/c2, was from KL → π+π−π0(D). At the KL mass region, the background was

from KL → π+π−γ, under the clear peak coming from the signal, KL → π+π−e+e−. In spite of

their small contributions, the Ξ decay and the simultaneous double decays were still populated in

higher mass tail.

We made further background reductions described in the following sections.

Mππee(GeV)

Eve

nts/

0.00

2(G

eV)

1

10

10 2

10 3

10 4

0.4 0.425 0.45 0.475 0.5 0.525 0.55 0.575 0.6

Figure 5.1: Mass distribution after the basic cut. To clarify the background contribution and the

signal, transverse momentum squared was restricted to be less than 0.0001 GeV2/c2.

52

5.3.1 MK Invariant Mass Cut

First, we defined the invariant mass window for four charged tracks. The window width was

|Mπ+π−e+e−−M0K | < 0.005GeV/c2, where MK0 = 0.4978GeV/c2 [9]. As described in Section 5.3.7,

this width corresponded to 2.9σ of the signal distribution.

5.3.2 Total Momentum Cut

The Ξ was produced at the target and had higher momentum of around 220 GeV/c at the peak.

This is a striking contrast to the KL, whose average momentum was 70 GeV/c. Figure 5.2 shows

the momentum distributions of the signal and the hyperon decays. To reject Ξ → Λ(→ pπ−)π0D,

we required the total momentum to be below 200 GeV/c. As shown in Figure 5.3, we can reject

72 % of the cascade decays, while keeping 99.6% of the signal.

5.3.3 Vertex χ2 Cut

For Ξ → Λ(→ pπ−)π0D, since the Λ has a finite lifetime(cτ = 7.89 cm), the vertex position of

Λ → pπ− has a certain offset from the primary vertex of Ξ → Λπ0D, which can be detected with

the KTeV spectrometer. Similarly, in the case of Ke3 simultaneous double decays, both Ke3’s

decay independently so that the vertex positions of them generally have a disparity. If we try

to reconstruct them with a common vertex, the quality of the vertex reconstructed from these

backgrounds were expected to be poor.

For example, let us define (xa2, zxa2) as the space point in the X view at DC2 for track a,

and σxa2 as the uncertainty on xa2, due to the chamber resolution and multiple scattering at

the upstream materials. Let us assume that tracks are straight lines coming from the vertex,

(xv, yv, zv), with slopes, sxi and syi, in x and y views, respectively. We can then form the vertex

χ2 using the distance between the straight lines and the chamber hits, as:

χ2 =4tracks∑

i

DC1, 2∑n

[(xin − xv) − sxi(zxin − zv)]2

σ2xin

+[(yin − yv) − syi(zyin − zv)]2

σ2yin

. (5.1)

Here we summed over four tracks, and DC1 and DC2.

Figure 5.4 shows vertex χ2 distribution for KL → π+π−e+e−, Ξ → Λ(→ pπ−)π0D, and Ke3

simultaneous double decay generated with the Monte Carlo simulation. The signal has a sharp peak

around the origin, while backgrounds do not. In order to reject Ξ and double Ke3 backgrounds,

the vertex χ2 was required to be less than 50. After all the previous cuts, the signal acceptance

was 98.9 % with the vertex χ2 cut, while the acceptances of Ξs and the Ke3 double decays were

35.8 % and 36.0 %, respectively.

5.3.4 Mee Cut

For the conversion backgrounds, such as KL → π+π−π0(γ), the invariant mass of electron-positron

pair originated from a real photon is smaller than that from a virtual photon in KL → π+π−e+e−.

53

Accepted

Pππee(GeV)

Arb

itrar

y U

nit

0

1000

2000

3000

4000

5000

6000

7000

8000

0 50 100 150 200 250 300 350 400

Figure 5.2: Total momentum distribution of

four charged tracks, estimated with the Monte

Carlo simulation. Note that each distribution

is plotted with an arbitrary scale. The decay

Ξ → Λ(→ pπ−)π0D dominated in the momen-

tum region above 200 GeV/c, while the sig-

nal was in lower momentum region. A small

peak around 60 GeV/c was an effect of the

interaction at materials such as the defining

collimeter. The Ke3 simultaneous double de-

cays also have higher total momentum than

the signal. After introducing a total momen-

tum constraint of less than 200 GeV/c, 72 % of

the cascade decay was rejected, whereas 99.6%

of the signal was kept.

Pππee(GeV)

Eve

nts/

4.0(

GeV

)

Accepted

1

10

10 2

10 3

10 4

0 50 100 150 200 250 300 350 400

Figure 5.3: Total momentum distribution

from each background source. Over 200

GeV/c of total momentum, the cascade decay

was dominated.

54

10

10 2

10 3

10 4

0 20 40 60 80 100 120 140 160 180 200

Vertex χ2

Accepted

Figure 5.4: Vertex χ2 distribution for KL → π+π−e+e−, Ξ → Λ(→ pπ−)π0D, and Ke3 simultaneous

double decay generated with the Monte Carlo simulation after basic cuts. The number of events

for the decays are not normalized. The signal has a sharp peak at the origin while backgrounds

do not. The cut for the vertex χ2 was introduced at 50. After all the previous cuts, the signal

acceptance was 98.9 % with this cut, while the acceptances of Ξs and the Ke3 double decays were

35.8 % and 36.0 %, respectively.

55

This means that the electron-positron pair have a very small opening angle. This is a prominent

feature to distinguish a virtual photon conversion from a real photon conversion.

Figure 5.5 shows Mee distributions for KL → π+π−e+e− and KL → π+π−π0 generated with

the Monte Carlo simulation. The conversion background is populated in a very low invariant

mass region while the signal has a very long tail. The cut at Mee > 0.002GeV/c2 gave a very clear

separation from the conversion background. This cut rejected 96.4% of KL → π+π−π0 background

and 97.7% of KL → π+π−γ background, while keeping 86.6 % of the signal.

In addition, even in the signal mode, the electron-positron pair has a very small angle because

they are the virtual photon conversion product. This implies that at the upstream end of the

spectrometer, the two tracks are still close together, so DC1 has to identify those close hits.

Because simultaneous hits in the same drift cell aggravated the precision of the tracking, the

Mee > 0.002GeV/c2 cut assured the track separation at DC1 is greater than 0.7 cm, equivalent to

one drift cell separation for the decay within the fiducial region.

Mee(GeV)

Arb

itrar

y U

nit

200

300

400

500

600700800900

1000

2000

3000

4000

5000

6000700080009000

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

Accepted

Figure 5.5: Mee distribution after all the pre-

vious cuts for KL → π+π−e+e− and KL →π+π−π0 generated with the Monte Carlo sim-

ulation.

Mee (GeV)

Eve

nts/

0.00

02(G

eV)

Accepted

10 3

10 4

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

Figure 5.6: Mee distribution from each back-

ground source. The conversion backgrounds

such as KL → π+π−γ and KL → π+π−π0

were populated in the region of Mee <

0.002GeV/c2.

5.3.5 p2t Cut

The total momentum of decay products must be the same as that of the decay particle. For this

fixed target experiment, it is useful to define p2t , the transverse momentum squared of the decay

products, relative to the straight line between the target and decay vertex.

Generally speaking, the p2t is exactly zero for KL → π+π−e+e−. However, p2

t has a a uniform

56

distribution for the decay with missing particles as shown in Figure 5.7. In our analysis, we use

this difference to reject the background, which mimicked the signal with some missing particles.

A constraint on p2t will be determined in the following section.

Accepted

Pt2(GeV2/c2)

Arb

itrar

y U

nit

0

2000

4000

6000

8000

10000

12000

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2x 10

-3

Figure 5.7: p2t distribution for KL →

π+π−e+e− and KL → π+π−π0D. The p2

t

distribution for the signal has a sharp peak

at the origin while it is almost uniform for

KL → π+π−π0D. We introduced a cut at

0.00004GeV2/c2 to separate the background

contribution as discussed in the following sec-

tion.

pt2 (GeV2)

Eve

nts/

0.00

0001

(GeV

2 )

Accepted

0

500

1000

1500

2000

2500

3000

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1x 10

-3

Figure 5.8: pt2 distribution from each back-

ground source. Most of residual background

in this plot was from KL → π+π−π0D.

5.3.6 Pp0kine Cut

After all the above cuts, the remaining background to KL → π+π−e+e− came from KL → π+π−π0D

decays where an extra photon was not detected. To reduce this background, we used a kinematic

variable called “Pp0kine” [51, 52](Appendix B).

The parameter Pp0kine shows a very unique feature of the kinematic relation in the mode

where one particle in the final state was either missed or ignored. In our case, we assumed

that the event is KL → π+π−π0 and we missed the π0 in its final state. The π0 longitudinal

momentum is reconstructed from the given information, π0 mass, π± momenta and their mass,

and the KL direction vector extrapolating from the target to the vertex position. We then look

at the square of π0 longitudinal momentum in the frame boosted along the KL flight direction,

where the longitudinal momentum of the π+π− system is equal to zero. This is the definition of

Pp0kine.

57

The exact formulation is as follows:

Pp0kine =(M2

K − M2π0 − M2

ππ)2 − 4M 2π0M2

ππ − 4M 2K(P 2

T )ππ

4[(P 2T )ππ + M2

ππ](5.2)

where Mππ is the invariant mass of the π+π− system, Mπ0 is the mass of π0, MK is the mass of

KL, and (P 2T )ππ is the square of the transverse momentum of the π+π− system with respect to

the kaon flight vector. Figure 5.9 shows a schematic view of the situation.

Target KL Flight DirectionVertex

π0

Pπ−Pπ+

Pp0kine

(PT)ππ

Figure 5.9: Schematic view of Pp0kine. Pp0kine is a longitudinal momentum squared of π0 in the

kaon rest frame reconstructed with KL flight direction and momenta of two charged pions.

The Monte Carlo predictions of the distribution of Pp0kine for the signal and KL → π+π−π0D

background are shown in Figure 5.10. For KL → π+π−π0D, Pp0kine is positive-definite because

the momentum of π0 must be a real. On the other hand, Pp0kine of the signal event does not

always take a positive value.

We will determine a constraint of the combination of p2t and Pp0kine in the next section.

5.3.7 Optimization of p2t and Pp0kine Cuts

Based on the Monte Carlo simulation, optimum cuts for p2t and Pp0kine are considered by looking

at a matrix in Table 5.1, with an invariant mass window defined in section 5.3.1. Since the

background is expected be dominated by KL → π+π−π0D, the number of background events was

estimated by KL → π+π−π0D Monte Carlo simulation.

In principle, there are three constraints to determine the optimum cuts configuration:

1. At least 1000 signal events are required to evaluate the theoretical predictions in the angular

asymmetry measurement with a good confidence level(3σ or larger).

2. Number of background events is small enough to ignore its effect in the form factor and

asymmetry measurements. Since we plan to include the background effect into systematic

58

Accepted

Pp0kine(GeV2/c2)

Arb

itrar

y U

nit

0

2000

4000

6000

8000

10000

12000

14000

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04

Figure 5.10: Pp0kine distribution for the sig-

nal KL → π+π−e+e− and the background

KL → π+π−π0D after basic cuts. The back-

ground is populated mostly in the positive,

physically allowed region, while the signal can

be in negative region. A small smearing into

the negative region of KL → π+π−π0D is due

to resolution effects.

Pp0kine (GeV2)

Eve

nts/

0.00

1(G

eV2 )

Accepted

1

10

10 2

10 3

10 4

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04

Figure 5.11: Pp0kine distribution from each

background source. In the negative-definite

region, an excess over the background was ex-

pected to be KL → π+π−e+e− signal.

59

uncertainty, number of background events should be less than 1/3 of the uncertainty from

the statistical fluctuation of the signal.

3. Since Pp0kine cut directly changes Mππ distributions in KL → π+π−e+e− decay, this cut

may affect the acceptance calculation. Therefore, Pp0kine cut should be loose.

From constraint 1, we should pick set of constraints with more than 1000 signal events. The

constraint 2 requires that the number of background should be less than 10 events if we assume

1000 signal events. If we require the constraint 3, only a set of cuts of p2t < 0.00004GeV2/c2 and

Pp0kine < −0.0025GeV2/c2 meets all of our constraints explained above. Therefore, we obtained

1173 events in the signal region and 9.03 expected background events. The final acceptance of the

signal was 1.392 % for the signal events, estimated with the KL → π+π−e+e− Monte Carlo.

After all the cuts, the signal window was expected to be dominated by the KL → π+π−e+e−

signal. In Figure 5.12, a sharp peak from the signal is shown while the events in the lower mass tail

were mostly from KL → π+π−π0D decays. The window width |Mπ+π−e+e− −M0

K | < 0.005GeV/c2

corresponds to the ±2.9σ of the signal distribution.

hhhhhhhhhhhhhhhhhhhpp0kine(GeV2/c2)

p2t (10−5GeV2/c2)

< 20 < 15 < 10 < 8 < 6 < 4 < 2

N/A 1813199.4

1778188.6

1731169.2

1679162.5

1620148.6

1485124.7

119188.5

< −0.00 149863.1

146757.86

142846.1

138741.1

134135.3

123230.4

98218.9

< −0.025 141435.5

138731.3

135219.5

131515.6

127412.5

11739.03

9394.51

< −0.05 138030.0

135525.8

132215.0

128712.5

124610.5

11508.00

9214.51

< −0.075 133025.4

130621.2

127410.5

12408.72

12027.70

11085.23

8874.51

< −0.100 126620.6

124216.4

12128.42

11806.68

11425.65

10513.19

8412.47

< −0.125 117316.7

115112.5

11225.95

10934.21

10573.19

9711.74

7761.02

Table 5.1: Signal to background ratio matrix table. (Numerator: number of events after the cuts.

Denominator: number of estimated background events from KL → π+π−π0D Monte Carlo.)

5.4 Acceptance and Data After Final Cuts

The previous section described the event selection to suppress backgrounds to a sufficient level.

Here, we turn to summarize the signal acceptance in each analysis phase. Table 5.2 shows the

estimated signal acceptance at each analysis phase. With the basic cuts, we lost nearly 3/4 of

events: the requirements for four tracks hitting the CsI calorimeter with a shared vertex, and

identified two pions and two electrons, lost 66% of the signal. The other loss by the basic cuts was

due to the fiducial volume cuts and the trigger verification. At the end, the signal efficiency was

1.39 %, and from data, we accepted 1173 events as signal candidates.

60

Invariant Mass (GeV)

Eve

nts/

0.00

1(G

eV)

0

50

100

150

200

250

300

0.46 0.48 0.5 0.52 0.54

Figure 5.12: Invariant mass distribution after final cuts. The mass window is defined as

|Mπ+π−e+e− − M0K | < 0.005GeV/c2, which corresponds to the ±2.9σ of the signal distribu-

tion. In this plot, the number of events in the signal window is 1173 events while the expected

number of backgrounds is 9. The residual background in the signal region was mostly from

KL → π+π−π0D. The normalization of the signal Monte Carlo is based on the previous publi-

cation, BR(KL → π+π−e+e−) = 3.2× 10−7 [12].

61

Table 5.2: The estimated signal acceptance at each analysis phase from KL → π+π−e+e− Monte

Carlo.Analysis phase Efficiency

Level 1 trigger 12.8%

Level 2 trigger 9.08%

Level 3 trigger 8.40%

Basic Cuts 2.18%

0.493 < MK < 0.503GeV/c2 2.07%

Pππee < 200GeV/c 2.06%

Vertex χ2 <50 2.05%

Mee > 0.002GeV/c2 1.78%

p2t <0.00004 (GeV2/c2) 1.49%

Pp0kine <-0.0025(GeV2/c2) 1.392%

5.5 Background Estimation

Since background contaminating the signal region causes systematic uncertainty in the analyses,

we have to understand it, and certify that it is ignorable. In this chapter, we summarize the

background level from each background source with the flux-normalized Monte Carlo simulation.

In addition, we fit the sideband of signal region to estimate the background, and compare it with

the estimation from the Monte Carlo prediction.

5.5.1 Background Level from Each Source

Here, we estimate the background by using the Monte Carlo simulation except for the simultaneous

double decays. We generated the same number of events as the KL flux for KL → π+π−π0D

because of its huge statistics. The number of generated events with the Monte Carlo simulation

corresponded to three times the KL flux for the other background sources.

For Ke3 simultaneous double decays, the background level was evaluated from the real data.

Namely, we looked at another possible final state, π±π±e∓e∓ from the Ke3 double decays, and

calculated the number of events in the signal region by applying the analysis cuts explained above.

This obtained number of events was considered to be the same as the number of background events

from Ke3 double decays, (KL → π+e−ν) + (KL → π−e+ν).

Table 5.3 shows the efficiency of each background source and the estimated number of back-

ground events, which was normalized with the measured KL flux. The most prominent background

was from KL → π+π−π0D. Some backgrounds did not appear in the background estimation so that

we set the upper limit at the 90% confidence level.

We determined that the number of background events was 11.3± 3.5.

62

Table 5.3: The estimated background acceptance of each background source generated with the

Monte Carlo. The estimated background levels were normalized to the KL flux, so the sum of

them is the final number of background events in the signal region.

Background source Efficiency Background events

KL → π+π−π0D 2.4 × 10−8 9.0 ± 3.0

KL → π+π−π0 < 1.2 × 10−11 < 0.8(90%C.L.)

KL → π+π−π0DD 2.7 × 10−7 0.3 ± 0.2

KL → π+π−γ < 3.0× 10−8 < 0.7(90%C.L.)

Ke3 double decay 2.0 ± 1.4

Ξ → Λπ0D < 8.3× 10−9 < 0.7(90%C.L.)

Total 11.3± 3.5

5.5.2 Background estimation with sideband fit

Since the background was mostly from KL → π+π−π0D, the background level was also estimated

by fitting the sideband with the distribution of KL → π+π−π0D Monte Carlo, in the region of

0.45 < Mππee < 0.48GeV/c2 after the final cut. The estimated number of the background events

in the signal region by this method was 10.6±3.3 events, consistent with the background estimation

from the flux normalization.

5.5.3 Summary of Background Level

The number of background events estimated by the flux normalization and sideband fitting indi-

cated approximately 11 events. This gives the signal to background ratio of over 100. Comparing

to the statistical uncertainty of the signal, 1173±34.2 events, the background was small enough to

be ignorable in both the form factor measurement and angular asymmetry analysis. Hence, in the

following chapter, the contribution from the background was treated as a systematic uncertainty,

and not subtracted.

5.6 KL Flux Calculation

In the previous section, we utilized an estimated KL flux in the collected data to normalize the

Monte Carlo event generation for background. The measured KL flux was used for signal normal-

ization, to optimize the cuts to suppress background, and to estimate the background level in the

signal region.

Since the normalization mode was required to have a similar feature as the signal to reduce

a systematic uncertainty, we used the decay KL → π+π−π0D for the normalization in our study.

KL → π+π−π0D has four charged tracks, including two electrons from internal conversion of a

virtual photon, two charged pions, and an extra photon.

63

Here, we explain the KL flux measurement in our analysis.

5.6.1 Acceptance and Flux Calculation

We defined the KL flux as the number of KL decays in the fiducial volume in the kaon energy range

of 20 to 220 GeV and decay position between 90 and 160 m downstream from the target. The

Monte Carlo KL → π+π−π0D decay was generated within the fiducial volume and energy defined

above.

The event selection for the normalization mode, KL → π+π−π0D, was performed by using the

exactly same tracking, clustering and vertexing algorithms as used for the signal mode.

The basic constraint was the same as for the signal described in Section 5.2 except for a

photon identification. The same fiducial volume cut was made for a vertex position shared with

four tracks. In addition, all four tracks were required to hit the CsI calorimeter, to identify two

electrons and two pions, which are consistent with KL → π+π−π0D. Only one extra photon was

required in the calorimeter with at least 2 GeV energy deposit. The extra photon was required to

be more than 0.4 m away from pion energy clusters, in order to avoid misidentification from pion’s

satellite clusters. The identified photon cluster and the two electrons were required to form the

π0 invariant mass; 0.127 < Me+e−γ < 0.143GeV/c2. Finally, KL invariant mass cut was made as

0.492 < Mπ+π−e+e−γ < 0.504GeV/c2. The summary of cuts is shown in Table 5.4.

Table 5.4: Cuts for K0L → π+π−π0

D.

Cut Description

95m < Zvertex < 154m Fiducial volume Cut

Photon Cluster Energy > 2.0GeV

Minimum Cluster Sep> 0.4m distance from π± to a photon

4 tracks hit the CsI Verify PID, Reject Hyperons

Vertex χ2 < 50 Ke3 Double decays and Hyperons

Charges consistent with π+π−π0D Ke3 Double decays

Mee > 0.002GeV/c2

0.127 < Meeγ < 0.143GeV/c2

0.490 < Mππeeγ < 0.506GeV/c2

The possible background source in the normalization mode was KL → π+π−π0 with a photon

converted at the vacuum window, since they had the same topology as four charged tracks and

a photon. The reduction for this background source was mostly done by invariant mass cut for

two electrons, Mee > 0.002GeV/c2, as described in the signal selection. Other background source

was the Ξ decay, but expected to be a very small contribution. The result of background study is

shown in Table 5.5. The background level itself, which was evaluated to be only 0.43% of the flux

of normalization mode, was found to be smaller than the other systematic uncertainties.

64

The acceptance of normalization mode was calculated based on the ratio of the number of

accepted Monte Carlo events to the number of generated events. The result was (0.6065±0.0005)%

for the winter run and (0.5850± 0.0005)% for the summer run, where errors are statistical only.

The number of accepted events was about 2.12 millions after subtracting 16000 background

events shown in Table 5.5. Using the branching ratio, Br(KL → π+π−π0D) = (0.151±0.0047)% [9],

the KL flux was estimated as (2.343± 0.0016)× 1011 KL decays.

Figure 5.13 shows the invariant mass distributions of data and the Monte Carlo simulation

after all cuts were applied. The discrepancy at higher mass region between data and Monte Carlo

simulation was of 0.3%, as described later.

0

200

400

600

800

1000

1200

1400

1600

1800

x 102

0.49 0.492 0.494 0.496 0.498 0.5 0.502 0.504 0.506

Data/MC ratio M (GeV)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.49 0.492 0.494 0.496 0.498 0.5 0.502 0.504 0.506

ππeeγ

M (GeV)ππeeγ

Figure 5.13: Distribution of mass of π+π−e+e−γ after all cuts. Dots shows data and histogram

shows the Monte Carlo simulation in upper plot. Lower plot shows ratios between data and the

Monte Carlo. The discrepancy at higher mass tail gave the flux uncertainty of only 0.3% level.

65

5.6.2 Systematic Uncertainty

We now consider systematic uncertainty in the normalization.

KL → π+π−π0D Branching Ratio

Particle Data Group [9] reports KL → π+π−π0 branching ratio as (12.56± 0.20)%, and π0 Dalitz

decay branching ratio as (1.198± 0.032)%. These cause the relative flux uncertainty of 1.6% and

2.7% respectively, and the quadratically combined error is 3.14%.

π0 Form Factor

Another possible error source was an uncertainty of the π0 form factor in the Dalitz decay. We

parameterized the form factor as

F = 1 + ax (5.3)

with a = 0.032 ± 0.004 for the slope [9], where x = (mee/mπ0)2. The form factor causes the

enhancement at high mass tail of Mee. This uncertainty to the flux measurement was very small

as expected from Equation 5.3. The error was evaluated as 0.046%, so the error could be neglected

in the flux measurement.

Background Subtraction

As shown in Table 5.5, since the fraction of background was as small as 0.43% of the KL →π+π−π0

D decays, the expected uncertainty from background subtraction was only 0.0019%. Thus,

we concluded that the uncertainty from background subtraction was negligible.

Table 5.5: Summary of the background level at the flux measurement.

Source Expected BG events

KL → π+π−π0 (16001± 75) events

Ξ → Λπ0D (1.5 ± 0.8) events

Ke3 Double decays (62± 7.9) events

Drift Chamber Inefficiency

A tracking inefficiency was found in neutral beam region of upstream drift chambers. The detail

will be described later in each signal analysis, so we give a short explanation and systematic study

for the normalization mode here.

The discrepancy between data and the detector simulation in the total tracking inefficiency in

upstream chambers was reported to be 5% in the worst case. To study the effect, we reduced the

tracking efficiency by 5 % in the beam region in the Monte Carlo simulation. This decreased the

66

acceptance of normalization mode by 1.85 %. Hence, we included this in systematic uncertainty.

However, this error is almost canceled out when taking the ratio in the branching ratio measurement

of KL → π+π−e+e−, so we will study this effect again in the branching ratio measurement.

Energy Cluster Threshold

For clustering of photons and electrons, we utilized the hardware cluster counting(HCC). The

threshold for the crystal at the local maximum of the cluster was approximately 1GeV. However,

the threshold level of each crystal in the Monte Carlo did not agree well with data. Figure 5.14

shows data and Monte Carlo comparison of HCC threshold distributions of 3100 CsI crystals for

both the winter and summer run. Obviously, the threshold level of Monte Carlo was higher by

about 5% in the winter run and 10% in the summer run, than the actual data. This effect in

the Monte Carlo could decrease the trigger efficiency of HCC requirement, especially for the low

energy electrons and photons.

In order to evaluate this effect, we artificially shifted the threshold for all crystals to agree with

data. This only changed 0.1 % of the number of Monte Carlo events. This is because that electrons

were required to solely satisfy Etotal and hardware cluster threshold at the trigger verification:

sum of two electron energies should be 11GeV or more, and each electron must have a momentum

of 2 GeV/c or higher.

Therefore, we assigned 0.1 % as the systematic uncertainty from this source.

0

200

400

600

800

1000

1200

0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

100

200

300

400

500

600

700

800

900

0.6 0.8 1 1.2 1.4 1.6 1.8 2

(GeV)

(GeV)

Data

Data

Monte Carlo

Monte Carlo

Figure 5.14: HCC threshold distribution. Solid histograms are from data, dashed ones from Monte

Carlo. Top plot is taken from the winter data and bottom is from the summer data. The peaks of

data were shifted by about 5 to 10%, compared to the Monte Carlo.

67

Mass Resolution

As shown in Figure 5.13, the comparison of Mππeeγ distributions between data and the Monte

Carlo was almost consistent. However, there was a small discrepancy at higher and lower mass

regions. This came from a photon reconstruction accuracy, such as the accidental overlay and

pion shower simulation. To see the effect of discrepancy, we changed the mass cut position to

0.492 < Mππeeγ < 0.500GeV/c2 and compared it with the sample with nominal cut. Applying

this mass cut, we found that the KL flux was changed by 0.4% for the winter run and 0.3% for

the summer run. We introduced an averaged error of 0.3% as a systematic uncertainty from this

source.

5.6.3 Summary

We summarize the flux measurement with the decay KL → π+π−π0D. Most part of uncertainty in

this measurement came from the branching ratio measurement of this mode, 3.14%, and the total

uncertainty is 3.7%, as shown in Table 5.6. Total KL flux was (2.343±0.002(stat)±0.085(syst))×1011, as shown in Table 5.7. Since the statistical error was 0.07% level in the flux calculation while

the error from the branching ratio uncertainty was approximately 3% level, the error was naturally

systematic dominant. As described above, some systematic sources will be canceled out when we

proceed to the branching ratio measurement.

Table 5.6: Summary of systematic uncertainty in the flux measurement.

Source Uncertainty

KL → π+π−π0D Branching ratio measurement 3.14%

Drift chamber inefficiency 1.9%

Mππeeγ mass resolution 0.3%

Cluster energy threshold 0.1%

KL → π+π−π0D Monte Carlo statistics 0.06%

π0 form factor 0.05%

Background subtraction 0.002%

Total 3.66 %

68

Table 5.7: Summary of the flux calculation with K0L → π+π−π0

D. Expected background was

already subtracted.

Dataset # Events(Data) KL flux

Winter 1301650 1.421× 1011

Summer 814227 0.922× 1011

Total 2115877 (2.343± 0.002(stat.)± 0.085(syst.))× 10 11

69

Chapter 6

Form Factor Measurement

In the decay matrix element of KL → π+π−e+e−, there is an ambiguity in the form factor mea-

surement for the gM1 magnetic dipole transition. Recently, KTeV collaboration has reported a

preliminary result on this measurement through a photon spectrum in the decay KL → π+π−γ[50].

Since we can make a precise measurement of the form factor through KL → π+π−e+e−, we in-

dependently determine the form factor here and compare distributions of characteristic variables,

especially the pions invariant mass, Mππ.

6.1 Formulation

The matrix element of KL → π+π−e+e− described in Chapter 1 matched the data well in most

of the kinematic variable distributions. However, Mππ distribution for the data shifted higher

than the Monte Carlo with a constant gM1, as shown in Figure 6.1. This is phenomenologically

understood as an effect of vector-meson dominance in the M1 direct emission amplitude. This

suggests that the photoemission in this kind of decays is also intermediated by a vector meson, ρ,

as shown in Figure 6.2.

This effect was already pointed out in Ref [22, 29] and observed through KL → π+π−γ in

Ref. [37]. The data and Monte Carlo comparison with and without the form factor in KL → π+π−γ

are shown in Figure 6.3. The effect makes the structure function of the form of

gM1 −→ gM1 · F,

F = a1[(M2ρ − M2

K) + 2MKEγ ]−1 + a2, (6.1)

where Mρ is the mass of ρ vector meson, MK is the mass of kaon, and Eγ is an energy of the

photon in the KL rest frame.

For the decay KL → π+π−e+e−, it was natural to modify the form factor by replacing the

photon in the decay KL → π+π−γ by a virtual photon which converts to e+e−. We used a

70

Mππ(GeV)

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100

200

300

400

500

0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48

Figure 6.1: Mππ distributions of data(dots) and Monte Carlo with a constant gM1(histogram) in

the decay KL → π+π−e+e−. The distribution of the Monte Carlo clearly shifted lower than that

of data.

71

KLη, η'

π+

π−

ρ γX

a)

KL

π+

π−

ω γXπ0

KLη, η'

π+

π−

ρ

γ

X KL

π+

π−γ

Xπ0

b)

ω

ρ

ρ

Figure 6.2: Contributions to the vector-meson intermediate model of the direct emission in the

decay KL → π+π−γ: (a) contact terms and (b) pole(ρ propagator) terms [29].

72

Figure 6.3: The gM1 form factor effect in the decay KL → π+π−γ. Eγ distribution with and

without a form factor. a)With a form factor taking account of vector meson intermediate. b)With

a constant gM1.

structure function:

F = a1[(M2ρ − M2

K) + 2MKEγ∗]−1 + a2, (6.2)

where Eγ in Equation 6.1 is substituted by the virtual photon energy, Eγ∗ = Ee+ + Ee− .

The precise measurement of the form factor will allow us to determine the angular asymmetry

and branching ratio accurately.

6.2 Maximum Likelihood Fit

In this section, we extract the constants in the form factor term, a1/a2 and a1 by fitting the data

event by event, using five independent variables, φ, Mππ, Mee, cos θπ+ , and cos θe+ . The definition

of the likelihood function with floating variables (a1/a2, a1) is

ln L(a1/a2, a1) =∑

i

lndΓ(φi, Mππ i, Mee i, cos θπ+ i, cos θe+ i; a1/a2, a1)/dφdMππdMeed cos θπ+d cos θe+

(Averaged Acceptance(a1/a2, a1)),

(6.3)

where dΓ(φi, Mππ i, Mee i, cos θπ+ i, cos θe+ i; a1/a2, a1)/dφdMππdMeed cos θπ+d cos θe+ is a differen-

tial cross-section for the set of kinematical variables (φi, Mππ i, Mee i, cos θπ+ i, cos θe+ i) for each

event calculated with the matrix element, Equation 1.23, described in Chapter 1. “Averaged Ac-

ceptance” is an averaged overall acceptance of KL → π+π−e+e− decay in this analysis. This

73

acceptance is a function of (a1/a2, a1), which depends on the detector, trigger, and analysis cuts.

The detailed formulation and fitting procedure are explained in Appendix A. Table 6.1 lists the

fixed physics parameters in the fitting.

Table 6.1: Fixed physics parameters in the maximum likelihood fit. Values are reported by the

Particle Data Group(PDG) [9] and the theoretical prediction [22].

Physics parameters values

η+− 2.285× 10−3

Φ+− 43.5

gE1 0.038

gP 0.15

The fit used the 1173 signal events described in the event selection. In addition, we prepared

0.5 million Monte Carlo events of KL → π+π−e+e− with the parameters of a1/a2 = −0.70 and

a1 = 1.0 after all the cuts, in order to calculate the “averaged acceptance” in Equation 6.3. For

different a1/a2 and a1, the “averaged acceptance” was reweighted accordingly. The maximization

of the likelihood function and the error estimation were performed with MIGRAD and MINOS

subroutines in MINUIT package in CERN program library [49], and the form factor was evaluated

by floating a1/a2 and a1 simultaneously.

Figure 6.4 shows a contour plot of the result from the maximum likelihood fit. The best values

were determined as

a1/a2 = −0.684+0.031−0.043(stat.) (6.4)

a1 = 1.05± 0.14(stat.) (6.5)

with statistical errors only.

6.3 Data and Monte Carlo Comparison

With the new form factor, we regenerated the Monte Carlo KL → π+π−e+e− events and compared

kinematical parameter distributions with the data. The distributions are shown in Figure 6.5.

In general, data and the Monte Carlo agree well, and we concluded that the new form factor

constants can reproduce the signal. Especially, the new form factor improves the agreement in

Mππ distribution between data and Monte Carlo simulation as shown in Figure 6.6.

6.4 Error on the Form Factor

We will next describe the systematic uncertainties on the form factor measurement. In this form

factor measurement, the systematic uncertainties may arise from four points: 1) fitting procedure,

2) theoretical ambiguity, 3) understandings of the detectors, and 4) residual background.

74

a1/a2

a 1

1σ

2σ3σ

a1/a2 = -0.684+0.031

-0.043

a1 = 1.05+0.14

-0.14

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

-0.75 -0.7 -0.65 -0.6 -0.55 -0.5

Figure 6.4: Fit result of a1/a2 and a1 from two dimensional maximum likelihood method. The

errors were statistical only. The statistical errors were determined with widths of the 1σ contour

projection on the axes of a1/a2 and a1 respectively.

75

Mππ Distribution, Data vs MCMππ Distribution, Data vs MCMππ Distribution, Data vs MC φ(rad)

0

20

40

60

80

100

120

140

-3 -2 -1 0 1 2 3

Mππ Distribution, Data vs MCMππ Distribution, Data vs MCMππ Distribution, Data vs MC Mππ(GeV)

0

20

40

60

80

100

120

140

0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48

Mee Distribution, Data vs MCMee Distribution, Data vs MCMee Distribution, Data vs MC Mee(GeV)

0

20

40

60

80

100

120

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022

cosθπ+ Distribution, Data vs MCcosθπ+ Distribution, Data vs MCcosθπ+ Distribution, Data vs MC cosθπ+

0

20

40

60

80

100

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

cosθe+ Distribution, Data vs MCcosθe+ Distribution, Data vs MCcosθe+ Distribution, Data vs MC cosθe+

0

20

40

60

80

100

120

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Figure 6.5: Comparison of various distributions between data and Monte Carlo. Dots represent

data and solid histograms represent the Monte Carlo. These five kinematic parameters completely

define a topology of the decay KL → π+π−e+e−. All parameters from data agree well with the

Monte Carlo.76

Mππ(GeV)

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100

200

300

400

500

0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48

Mππ(GeV)

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0

100

200

300

400

500

0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48

Figure 6.6: Mππ distributions of data and Monte Carlo with and without the gM1 form factor in

the decay KL → π+π−e+e−. Left: a constant gM1. The distribution of the Monte Carlo is clearly

shifted lower than that of data. Right: gM1 with the new form factor. The data and Monte Carlo

agree well.

In the following sections, we will examine uncertainties from each source.

6.4.1 Fitting Procedure

The accurate form factor fitting is a critical part of this measurement. In order to find the

systematic error due to fitting, we examined the following:

1. Monte Carlo sample statistics in the “averaged acceptance” calculation in Equation 6.3,

2. Whether the fit gives the correct values,

3. Any possible bias caused by the fitting procedure.

In order to reduce the statistical uncertainty in each error estimation, we generated the Monte

Carlo events and used them as “pseudo data” for the fit, instead of the actual data.

First, we studied whether the 0.5 million Monte Carlo events we used to calculate the “averaged

acceptance” was sufficient or not. In order to study this, we calculated the “averaged acceptance”

with four different samples ranging from 0.05 million to 2 million events. These Monte Carlo events

were generated with the parameters of a1/a2 = −0.70 and a1 = 1.0. The fits were performed on the

actual 1173 signal events. The fit results for a1/a2 and a1 are shown in Table 6.2. In addition, in

order to look at the fit accuracy, besides the actual data, we fitted two sets of “ pseudo data”, which

77

were 25000 Monte Carlo signal events generated with the parameters of (a1/a2, a1) = (−0.70, 1.0)

and (−1.0, 0.70), respectively. These results are also shown in Table 6.2. Obviously, all the fit

results were well-converged and stable in the range of the 0.05 million through 2.0 million Monte

Carlo sample events in the “acceptance calculation.” Therefore, we conclude that the 0.5 million

sample events in the “acceptance calculation” was sufficient. We assigned the largest deviations

of the fit results from the results with 0.5 million sample events in Table 6.2 as the systematic

uncertainties; 0.002 for a1/a2 and 0.01 for a1.

Table 6.2: Sample size dependency in the ”acceptance calculation.” This table shows the depen-

dency between the number of Monte Carlo sample events in the “acceptance calculation” and fit

results. The fitted result for each dataset was well-converged and stable.

Inputs for pseudo data (Actual data) a1/a2 = −0.70, a1 = 1.0 a1/a2 = −1.0, a1 = 0.70

Acc. Calc. a1/a2 a1 a1/a2 a1 a1/a2 a1

2.0 millions -0.684 1.06 -0.702 0.97 -1.09 0.75

0.5 millions -0.684 1.05 -0.703 0.97 -1.10 0.75

0.2 millions -0.682 1.06 -0.701 0.97 -1.09 0.75

0.05 millions -0.683 1.04 -0.703 0.96 -1.09 0.74

Next, we looked at whether the fitting procedure gives correct results. In this study, we gener-

ated sets of “pseudo data” of about 25000 events with parameters ranging −0.80 ≤ a 1/a2 ≤ −0.60

and 0.9 ≤ a1 ≤ 1.1, and then we looked at the fitting results for those datasets. The results are

shown in Figure 6.7. We conclude that the fitting procedure gives a correct result, in the range of

−0.80 ≤ a1/a2 ≤ −0.60 and 0.9 ≤ a1 ≤ 1.1.

We also looked for an intrinsic bias in the fitting procedure which may cause a systematic offset

of the parameters. To estimate this, we fitted ten independent “pseudo data,” generated with

a1/a2 = −0.70 and a1 = 1.0. Each sample had 0.2 million signal events, in order to reduce the

statistical uncertainty. The two-dimensional distribution of results is shown in Figure 6.8. Total

statistical uncertainties of the ten samples for a1/a2 and a1 are 0.0013 and 0.0044, respectively,

while the systematic shifts between fit results and input values are 0.0005 and 0.0026, respectively.

Therefore, we could not find any fitting bias within the statistical uncertainty. Since the bias was

limited by the statistical uncertainty in this study, we assigned systematic uncertainties for a1/a2

and a1 from this source as 0.0013 and 0.0044, respectively.

Thus, we assigned combined errors, 0.0024 for a1/a2, and 0.011 for a1, into systematic uncer-

tainties from the understandings of the fit procedure.

78

(Input a1)

a1/a

2 fit

res

ults

-0.85

-0.8

-0.75

-0.7

-0.65

-0.6

-0.55

-0.5

0.85 0.9 0.95 1 1.05 1.1 1.15

(Input a1/a2)

a1 fi

t res

ults

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

-0.825 -0.8 -0.775 -0.75 -0.725 -0.7 -0.675 -0.65 -0.625 -0.6 -0.575

Figure 6.7: Fit results of a1/a2(left plot) and a1(right plot) in the range of −0.80 ≤ a1/a2 ≤ −0.60

and 0.9 ≤ a1 ≤ 1.1. The fits gave correct results for any datasets.

6.4.2 Smearing by Detector Resolution

The smearing of any kinematic variables could occur since detectors, especially the spectrometer,

had a finite precision and inefficiency. This effect could cause the deviation in the momentum

and track hit position measurements(i.e., the decay topology), and give ambiguities in the fitting

procedure.

In principle, the matrix element in the numerator in Equation 6.3 should be calculated with

the true values of the five kinematic variables, whereas we could only obtain the reconstructed and

smeared values from data. This means that the difference between true and reconstructed(smeared)

values in the numerator may give some uncertainty.

In order to clarify this issue, we first fit a dataset of 25000 events generated by the Monte Carlo

simulation, by using reconstructed variables as usual. We then fit the same dataset by using the

true five variables, and looked at the difference between them.

Table 6.3: Smearing effect by detector resolution in determination of the form factor parameters.

The matrix element was calculated for the same Monte Carlo dataset by using both true and

reconstructed variables. The difference between the the true and reconstructed variables results

was small enough.

Fitted data a1/a2 = −0.70, a1 = 1.0 a1/a2 = −1.0, a1 = 0.70

a1/a2 a1 a1/a2 a1

True -0.703(16) 0.971(53) -1.10(11) 0.743(61)

Reconstructed -0.704(17) 0.969(53) -1.09(11) 0.743(61)

79

a1/a2

a1

0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

-0.71 -0.708 -0.706 -0.704 -0.702 -0.7 -0.698 -0.696 -0.694 -0.692 -0.69

Figure 6.8: The fit results of a1/a2 and a1 for the ten pseudo data samples including 0.2 million

events with a1/a2 = −0.70 and a1 = 1.0. Each data point represents the fit result of a1/a2 and

a1. The resulting statistical uncertainties are 0.0013 for a1/a2 and 0.0044 for a1, respectively. The

averages of fit results were -0.7005 for a1/a2 and 0.9974 for a1.

80

Table 6.3 shows the results of the fittings. In this study, we also examined two different sets of

parameters in fitted dataset. The difference between the true and reconstructed variables results

was very small. Hence, we conclude that the smearing effect by the detector resolution is negligible.

The errors from the smearing effect were determined to be 0.001 for a1/a2 and 0.002 for a1, around

the region of a1/a2 = −0.7 and a1 = 1.0.

6.4.3 Vertexing Quality

The understanding of the quality of vertex reconstruction was a critical issue to determine the

kinematics in KL → π+π−e+e−. As shown in Figure 6.9 for KL → π+π−e+e−, the distributions

of vertex χ2 of the data and Monte Carlo simulation did not agree well. This difference could cause

the ambiguity in the form factor measurement.

In order to estimate the uncertainty, we first compared the vertex χ2 distributions between data

and Monte Carlo simulation for KL → π+π−π0D decays. We then took the ratio between data and

the Monte Carlo in each vertex χ2 bin, and modified the event weight of KL → π+π−e+e− Monte

Carlo by the difference in the “averaged acceptance” calculation. The vertex χ2 distribution after

the modification is shown in Figure 6.10.

vertex chi2

datppee/mcppee ratio

1

10

10 2

0 5 10 15 20 25 30 35 40 45 50

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 5 10 15 20 25 30 35 40 45 50

Figure 6.9: Data and Monte Carlo comparison

in vertex χ2 for KL → π+π−e+e−. Top plot

shows the overlay of data(dots) and Monte

Carlo(histogram). Bottom shows the data to

Monte Carlo ratio. The same effect was ob-

served in the decay KL → π+π−π0D.

vertex chi2

datppee/mcppee ratio

1

10

10 2

0 5 10 15 20 25 30 35 40 45 50

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 5 10 15 20 25 30 35 40 45 50

Figure 6.10: Data and the modified Monte

Carlo comparison in vertex χ2 for KL →π+π−e+e−. The modification was performed

with event weights evaluated from the com-

parison between data and the Monte Carlo of

KL → π+π−π0D.

By using the modified weight, we obtained the fit result of a1/a2 = −0.686 ± 0.036 and a1 =

1.04 ± 0.14. Therefore, we assigned systematic errors of 0.3 % for a 1/a2 and 1.0 % for a1 to the

uncertainty in the understanding of vertex χ2 distribution.

81

6.4.4 Drift Chamber Inefficiency

As shown in Figure 6.11, the Mee distribution originated from a virtual photon was well simulated

in KL → π+π−π0D. This is a supporting evidence that the chamber response for closed two electron

tracks was well understood. However, although track illumination in the chamber matched between

data and the Monte Carlo in most of the regions, there is a small discrepancy in the neutral beam

regions of DC1 and DC2, as shown in Figure 6.12. This was caused by a surplus in higher side in

the drift chamber SOD distribution in data, which lead to extra loss. The higher tail is thought

to be caused by the radiation damage on the sense wires which lowered the gain.

In the form factor measurement, the inefficiency may cause the acceptance deficit in some

charged track configurations and change the kinematic parameters. To estimate uncertainty from

this source, the inefficiency for DC1 and DC2 was artificially added to the Monte Carlo simulation.

Adding 5% inefficiency changed a1/a2 and a1 by 0.3% and 0.6%, respectively. Hence, we determined

them as a systematic uncertainty in the tracking.

6.4.5 Momentum Scale

Momentum scale in the spectrometer was a critical part in the form factor measurement, since

the accurate momentum reconstruction of charged tracks directly changed the topology of KL →π+π−e+e−. This uncertainty could be caused from uncertainty in the detector alignment and

magnet field measurement.

In order to evaluate the momentum scale in the spectrometer, we looked at the difference in the

distribution of Mππee from KL → π+π−π0D between data and Monte Carlo, since the error in the

momentum measurement is proportional to the error in the measurement of (Mππee−2Mπ−2Me).

Figure 6.13 shows the invariant mass distributions of the four tracks for data and Monte Carlo

simulation. The difference between data and the MC was less than 0.01% in mean value of the

distributions.

In addition, Figure 6.14 shows the averaged Mππ as a function of (Eπ1+Eπ2) for KL → π+π−π0D

data and Monte Carlo simulation, in order to evaluate the momentum dependency. The error in

the momentum measurement is also proportional to the error of (Mππ − 2Mπ). A drop in Mππ

lower energy region is caused by the detector acceptance. The ratio was perfectly flat in the whole

region.

Therefore, we conclude that the fluctuation from the momentum scale was negligible.

6.4.6 Physics Input Parameter

In this study, the physics parameters in the matrix element calculation were taken from Ref. [9],

which have their own errors. The deviations of such parameters may vary the fit results. In order

to evaluate this effect, we produced “pseudo data” of 25000 events by independently varying the

physics parameter, η+−, Φ+−, gE1 and gP , within a reasonable range, and fitted those datasets.

For η+−, Φ+−, we independently varied the values within ±1σ, quoted in Ref. [9]. For gE1, there

82

Data/MC ratio

0

10000

20000

30000

40000

50000

60000

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Mee(GeV)

Mee(GeV)

Figure 6.11: Data and Monte Carlo comparison for Mee distributions in the decay KL → π+π−π0D.

Upper plot shows the Mee distributions of data(dots) and Monte Carlo(histogram) in the decay

KL → π+π−π0D. Lower plot shows the data to Monte Carlo ratio. The distribution of the Monte

Carlo agreed well with that of data in a whole distribution.

83

Data/MC ratio

0

10000

20000

30000

40000

50000

60000

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-0.6 -0.4 -0.2 0 0.2 0.4 0.6Y view(m)

Y view(m)

Figure 6.12: Data and Monte Carlo comparison of track illumination at DC1 in Y view. Upper plot

shows the comparison between data(dots) and Monte Carlo(histogram) of the track illumination.

Lower is the data to Monte Carlo ratio. There is small inefficiency in neutral beam region(|Y | <

0.05m).

84

data/mc ratio

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5

Mππee(GeV)

Mππee(GeV)

Figure 6.13: Distributions of Mππee from

KL → π+π−π0D of data and Monte Carlo. Up-

per plot is the distributions of Mππee for data

and Monte Carlo, lower is the ratio of them.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

10 20 30 40 50 60 70 80 90 100

Eπ1+Eπ2 (GeV)

Mππ

(GeV

)

Eπ1+Eπ2 (GeV)

DA

TA

/MC

(Mππ

)

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

10 20 30 40 50 60 70 80 90 100

Figure 6.14: Averaged Mππ as a function of

(Eπ1 + Eπ2) for KL → π+π−π0D data and

Monte Carlo simulation. Upper plot is aver-

aged Mππ distributions, and lower shows the

ratio. A drop in Mππ lower energy region is

caused by the detector acceptance.

is no experimental result and it is only theoretically predicted as gE1 < (0.15 × gM1). For gP ,

there is no experimental result on the charge radius measurement for K 0, and no valid calculation

on this. Therefore, we looked at fit results from the zero contributions of gE1 and gP , respectively.

Fitting results are shown in Table 6.4 with the variation range of the parameters. We quadrat-

ically summed those errors and included 6.4% for a1/a2 and 12.4 % for a1 as the systematic

uncertainty.

6.4.7 Background Uncertainty

In this study, we have assumed the background level is ignorable since the signal to background ratio

is expected to be about 100. To confirm this, we also tested the contribution from background

uncertainty. We made a dataset with 1173 Monte Carlo signal events, and added 0, 12, and

23 background events from KL → π+π−π0D Monte Carlo events surviving all the analysis cuts.

Table 6.5 shows the variation of the parameters obtained by the fit. The shift caused by the

background is comparable to the statistical error from the additional 12 events, 0.006 for a 1/a2,

and 0.05 for a1. Therefore, we include 0.006 for a1/a2 and 0.05 for a1 as systematic uncertainties.

85

Table 6.4: Stability to parameter fluctuations.

Variation a1/a2 a1

Input value -0.700 1.00

η+− + 1σ -0.714(17) 0.98(5)

η+− − 1σ -0.685(17) 1.03(5)

Φ+− + 1σ -0.705(17) 1.08(5)

Φ+− − 1σ -0.709(17) 0.98(5)

gE1 = 0.0 -0.692(17) 0.94(5)

gP = 0.0 -0.744(22) 0.87(5)

Table 6.5: Background uncertainty

Data set a1/a2 a1

Input value -0.700 1.00

Signal MC only -0.694(39) 1.05(14)

+12 BG events -0.688(38) 1.00(14)

+23 BG events -0.690(38) 0.97(14)

6.4.8 Analysis Dependencies

The cut dependency in any kinematic variables is often examined in the systematic study of high-

energy physics analysis. This is an easy way to look at the discrepancy between data and our

understanding of the detector and signal. Here, we exhibit the cut dependencies for the form

factor analysis in Table 6.6. We assigned errors of 4.2 % for a1/a2, and 11.7 % for a1, respectively.

6.4.9 Systematic Uncertainty: Summary

So far, we evaluated the systematic uncertainties from various sources. We summarize the system-

atic uncertainties in Table 6.7.

6.5 Form Factor Measurement: Summary

We conclude that the form factors for KL → π+π−e+e− are

a1/a2 = −0.684+0.031−0.043(stat.) ± 0.053(syst.)

a1 = 1.05± 0.14(stat.)± 0.18(syst.).

86

Table 6.6: Deviations of analysis cut dependencies.

Cut a1/a2 a1 Variation

Z vertex 0.6% 1.2% 110m < Z < 154m

Cluster Threshold 0.3% 2.4% Shift HCC threshold

Mee cut 3.8% 8.5% Mee > 2.0 – 4.0 MeV/c2

Mππee cut 1.2% 4.7% ±6.0 – ±14.0MeV/c2

Pp0kine cut 0.3% 3.8% −0.0025 – −0.075MeV2/c2

E/p cut 0.3% 0.9% ±0.06 – ±0.10

p2t cut 0.9% 4.5% p2

t < 30 – 60MeV2/c2

Total 4.2 % 11.7%

Table 6.7: Systematic uncertainties.

Sources a1/a2 a1

Fit procedure 0.35% 1.0 %

Detector smearing 0.1% 0.2 %

Vertex quality 0.3% 1.0 %

DC ineff. 0.3% 0.6 %

Input Param. 6.4% 12.4 %

BG subtraction 1.5% 4.8 %

Analysis dep. 4.2% 11.7 %

Total 7.8% 17.8 %

87

Chapter 7

Asymmetry Measurement

As described in Chapter 1, CP violation in KL → π+π−e+e− arises from the interference of the

CP conserving direct emission(DE) and CP violating inner bremsstrahlung(IB) of KL → π+π−γ∗,

resulting in the angular asymmetry between normals to π+π− and e+e− decay planes.

An asymmetry between the number of events observed in particular angle regions may differ

from the theoretically predicted asymmetry, because the measured asymmetry depends on the

detector, trigger and analysis acceptances. Therefore, the asymmetry should be corrected for

the acceptance. We will work on different levels of asymmetries in this thesis, so we clarify a

terminology of these asymmetries here.

• Intrinsic Asymmetry: True asymmetry defined by the nature. This is what we have pursued

to measure from data and compare to the theoretical prediction. The intrinsic asymmetry is

given a priori.

• Input Asymmetry: Input value of the asymmetry into the Monte Carlo event generator. In

principle, the input asymmetry is equivalent to the intrinsic asymmetry but it was given by

us.

• Raw Asymmetry: Measured asymmetry with the KTeV detector and certain analysis cuts.

The raw asymmetry is not equal to the intrinsic and input asymmetry since the detector

and analysis acceptance affect on the distributions of kinematic variables related to the

asymmetry. Therefore, the raw asymmetry is a KTeV- and analysis-specific variable.

• Acceptance Corrected Asymmetry: Asymmetry of reconstructed events after correcting for

the acceptance. The acceptance correction was done by the Monte Carlo simulation. The

acceptance corrected asymmetry is directly comparable to the intrinsic and input asymme-

tries.

In this chapter, we first explain the raw asymmetry measurement, and clarify its origin. Next,

we show how the acceptance correction was done to estimate the acceptance corrected asymmetry.

Finally, we summarize the systematic uncertainties in this study.

88

7.1 Raw Asymmetry

7.1.1 Raw Asymmetry Calculation

First we start by the definition of raw asymmetry in this asymmetry study. The raw asymmetry

was defined as a simple counting asymmetry of φ distribution. Figure 7.1 is φ distribution of data

and Monte Carlo, after applying all the cuts as explained in Chapter 5. Dots are from data while

a solid histogram is from KL → π+π−e+e−, with input asymmetry of 0.14. The raw asymmetry is

calculated by

Asym. =NI,III − NII,IV

NI,III + NII,IV, (7.1)

where Ni corresponds the number of events in I, II, III, and IV angular quadrants in the plot.

The asymmetry becomes more apparent when they are shown in 2 cosφ sin φ instead of φ, as shown

in Figure 7.2. Regions of I and III in Figure 7.1 correspond to sinφ cos φ > 0 region, while II

and IV correspond to sin φ cosφ < 0 region. Therefore, the raw asymmetry was evaluated from

the positive-negative asymmetry in sin φ cosφ distribution:

Asym. =Nsin φ cos φ>0 − Nsinφ cos φ<0

Nsin φ cos φ>0 + Nsinφ cos φ<0. (7.2)

The asymmetries defined here are clearly observed in Figure 7.1 and 7.2.

Of the 1173 events observed in our analysis(Chapter 5), 719 events were observed in sinφ cosφ >

0 region, and 454 events were observed in sin φ cosφ < 0 region. The resulting raw asymmetry

calculated with Equation 7.2 was:

Asym. = 0.237± 0.029(stat.), (7.3)

with statistical error only. Note that this asymmetry is not directly comparable to the theoretical

prediction, as mentioned before. For example, KL → π+π−e+e− Monte Carlo events generated

with the input asymmetry of 0.147 gave the raw asymmetry of 0.258. This implies that there is a

mechanism to enhance the asymmetry, or some systematic bias in the asymmetry measurement.

Hence, the origin of this discrepancy will be examined in the following sections. After this issue

is cleared, we will move on to the acceptance correction on this raw asymmetry to obtain the

acceptance corrected asymmetry.

7.1.2 Bias from Detector and Event Selection

Before proceeding to the acceptance correction for the raw asymmetry, we verify that the raw

asymmetry actually came from the intrinsic asymmetry in KL → π+π−e+e− kinematics, and not

from the detector alignment or analysis cuts. If the asymmetry is caused by detector misalignment

or analysis cuts, the asymmetry will also be possibly introduced to other decay modes with similar

decay topology.

89

φ(rad)

I II III IV

0

20

40

60

80

100

120

140

-3 -2 -1 0 1 2 3

Figure 7.1: φ distribution of data, overlaid

with the Monte Carlo. Dots are from data

while a histogram is from KL → π+π−e+e−

Monte Carlo with the input asymmetry of

0.14. Cuts were applied to data and Monte

Carlo, as explained in Chapter 5. The raw

asymmetry was calculated from (NI,III −NII,IV )/(NI,III + NII,IV ), where Ni corre-

sponds the number of events in quadrants

I, II, III, and IV .

2*sin(φ)cos(φ)

sinφcosφ = - sinφcosφ = +

0

50

100

150

200

250

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Figure 7.2: Distribution of (2 cosφ sin φ) from

data, overlaid with the Monte Carlo. Dots

are from data while a histogram is from

KL → π+π−e+e− Monte Carlo. The quad-

rants I and III in Figure 7.1 correspond to

2 sinφ cos φ > 0, while the quadrants II and

IV in Figure 7.1 correspond to 2 sinφ cosφ <

0. The asymmetry is more apparent in this

plot.

90

First, we examine the raw asymmetry in the decay KL → π+π−π0D, since its decay topology is

similar to the signal mode, and there is no correlation between decay planes of π+π− and e+e−

from π0D decay. Moreover, this mode had enough events to detect a small amount of the asymmetry

in the measurement. Figure 7.3 shows the φ distribution in the π+π−e+e− center-of-mass frame

from data sample of 1 million events, positively identified as KL → π+π−π0D. The raw asymmetry

of this mode was calculated to be −0.00018± 0.00051. It is consistent with zero raw asymmetry

with a very good precision. This means that any raw asymmetry induced by the detector or event

selection is negligible.

We also looked at the asymmetry for events inside or outside of the signal region in π+π−e+e−

invariant mass. As shown in Figure 5.12, the signal region(0.493 < Mππee < 0.503GeV/c2) and

adjacent regions(0.483 < Mππee < 0.493GeV/c2 and 0.503 < Mππee < 0.513GeV/c2) were domi-

nated by the signal, while the outside regions(Mππee < 0.483GeV/c2 and Mππee > 0.513GeV/c2)

were dominated by KL → π+π−π0D. Since the intrinsic asymmetry of KL → π+π−π0

D is consistent

with zero, we can also expect a finite raw asymmetry in the outside regions if analysis cuts induced

a raw asymmetry.

Table 7.1 and Figure 7.4 show the raw asymmetry for different invariant mass regions. In order

to increase number of events in the sideband region, an event sample without pp0kine cut was also

examined. For both samples with and without pp0kine cut, prominent raw asymmetries appear in

the signal region, whereas no significant raw asymmetry is found in the outside region away from

the kaon mass. We thus can conclude that the event selection did not induce the raw asymmetry.

From above studies, we conclude that both detector and event selection do not spontaneously

cause a finite raw asymmetry from samples with zero intrinsic asymmetry. In the next section, we

will support this argument by a Monte Carlo study.

Table 7.1: Raw asymmetries at different mass regions. In our study, signal region was defined

as 0.493 < Mππee < 0.503GeV/c2. To increase number of events in the sideband region, a event

sample without pp0kine cut was also examined. No significant asymmetry appeared in the outside

region, in contrast to the signal region and its adjacent region.

Invariant Mass Region(GeV/c2) Raw Asym. Raw Asym.(without pp0kine cut) Description

< 0.473 0.0066± 0.048 0.0027± 0.0051 outside

0.473 - 0.483 0.25± 0.15 0.0027± 0.031 outside

0.483 - 0.493 0.35± 0.11 0.077± 0.034 adjacent

0.493 - 0.503 0.237± 0.029 0.185± 0.025 signal

0.503 - 0.513 0.0± 0.21 0.15± 0.19 adjacent

> 0.513 −0.063± 0.18 0.0 ± 0.16 outside

91

Raw Asym.=-0.00018±0.00051KL → π+π-πD

0

φ(rad)

0

10000

20000

30000

40000

50000

60000

70000

80000

-3 -2 -1 0 1 2 3

Figure 7.3: φ distribution of identified KL →π+π−π0

D. The cuts for the normalization was

applied to collect events. The φ asymmetry

is equal to zero with a very good precision,

so that the detector-induced asymmetry was

negligible.

Outside Region

Inv. Mass(GeV)R

aw A

sym

.

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.4

53

-0.4

63

0.4

63

-0.4

73

0.4

73

-0.4

83

0.4

83

-0.4

93

0.4

93

-0.5

03

0.5

03

-0.5

13

0.5

13

Figure 7.4: Raw asymmetries at different

mass regions. In our study, the signal re-

gion was defined within 0.493 < Mππee <

0.503GeV/c2 while the outside regions were

defined as Mππee < 0.483GeV/c2 and

Mππee > 0.513GeV/c2. To increase number of

events in the sideband regions, a event sample

without pp0kine cut was also examined. No

significant asymmetry was found in the out-

side region, in contrast to the signal region.

In signal region, because of background con-

tamination, the rawasymmetry is diluted for

the event set without Pp0kine cut.

92

7.1.3 Origin of Raw Asymmetry

So far, we have considered a source of the raw asymmetry other than the intrinsic asymmetry, and

concluded that the raw asymmetry is not caused by systematic uncertainties such as the detector

misplacement or event selection. Here, we turn to look at the behavior of the raw asymmetry

and the reason why the asymmetry is enhanced by the measurement. We will use the Monte

Carlo simulation of KL → π+π−e+e−, since it can reveal the relation between the input and raw

asymmetry, and it allows us to do a detailed study on the kinematical acceptance.

First, a relation between the input and raw asymmetry is examined to identify the source of

such enhancement. Since the input asymmetry is a function of the relative phase between the inner

bremsstrahlung and M1 direct emission amplitudes, various Monte Carlo samples were generated

with input asymmetries ranging from zero to nominal(0.147) by changing the indirect CP violating

phase, Φ+−, from 0 to π/4.

Table 7.2 and Figure 7.5 show the relation between the input and raw asymmetries in the Monte

Carlo simulation. The raw asymmetry is nearly proportional to the input asymmetry. Therefore,

we can connect the input and raw asymmetry with one constant;

RawAsym. = 1.75× InputAsym.. (7.4)

The coefficient greater than 1 indicates the raw asymmetry is enhanced than the input asymmetry.

Also, since there is no offset to this linear relationship, it supports that the raw asymmetry is

produced by the input asymmetry.

Table 7.2: Comparison between the input and raw asymmetry. The all analysis cuts were applied

to Monte Carlo data samples. Errors are statistical only. The result means that the raw asymmetry

is an enhancement of the input asymmetry.

Input Asymmetry Raw Asymmetry

-0.048 −0.085± 0.0060

-0.00215 −0.00933± 0.0060

0.022 0.051± 0.0060

0.047 0.077± 0.0059

0.070 0.128± 0.0059

0.090 0.146± 0.0068

0.124 0.212± 0.0068

0.146 0.245± 0.0067

0.147 0.258± 0.0067

0.153 0.260± 0.0066

93

Input Asymmetry

Raw

Asy

mm

etry

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

-0.05 -0.025 0 0.025 0.05 0.075 0.1 0.125 0.15

Figure 7.5: Relation between the input and raw asymmetry. The raw asymmetry is almost propor-

tional to the input asymmetry.

94

Next, we explore why the raw asymmetry is enhanced than the input asymmetry.

The KTeV detector was not sensitive to low momentum charged particles, especially the mo-

mentum less than 2 GeV/c in the lab frame. This was because the analysis magnet in the spec-

trometer kicked out those low momentum charged particles. Such particles were mostly electrons

since electrons had relatively small momenta. In this analysis, we lost approximately 70% of signal

events by this reason.

Figure 7.6 shows the situation. The open histogram is a φ distribution of KL → π+π−e+e−

Monte Carlo, before any detector, trigger, or analysis cuts. The hatched histogram is the distribu-

tion of Monte Carlo events which has at least one low momentum electron with less than 2 GeV/c

in the lab frame. As mentioned, we lost the hatched part in the plot, and only the remainder

was accepted in this analysis. Since the lower part has a relatively small angular asymmetry, the

remaining has an enhanced asymmetry, 0.229 in this plot. This enhanced asymmetry is compa-

rable to the observed raw asymmetry. This explains the origin of the enhancement in the raw

asymmetry.

Here, we summarize the raw asymmetry enhancement study in this section:

• Measured raw asymmetry of data is 0.237 ± 0.029(stat.), while the Monte Carlo with the

input asymmetry of 0.147 gave the raw asymmetry of 0.258.

• The detector or analysis did not induce the raw asymmetry. This is also supported by the

signal Monte Carlo with zero input asymmetry.

• The input and raw asymmetry have the simple linear relation which suggests the asymmetry

enhancement.

• The KTeV detector only accepted events with high momentum electrons. Those events

showed the large asymmetry.

7.2 Acceptance Corrected Asymmetry

In this section, we describe the measurement of the acceptance corrected asymmetry. Since the

acceptance corrected asymmetry is evaluated by converting the raw asymmetry as described before,

we discuss the acceptance correction first. Then, the acceptance corrected asymmetry is examined.

In Section 7.1, we showed that the raw asymmetry is enhanced because the events with low mo-

menta which have a smaller asymmetry are lost by the detector acceptance. In actual measurement

of the data, we have to make the acceptance correction with a proper Monte Carlo simulation to

evaluate the acceptance corrected asymmetry. The acceptance correction was done in the following

steps:

95

0

5000

10000

15000

20000

25000

30000

35000

40000

-3 -2 -1 0 1 2 3

φ(rad)

Generated SignalsAsym.=0.147

Acceptable Signals(Ee>2GeV)Asym.=0.229

Lost Signals(Ee<2GeV)Asym.=0.115

Figure 7.6: Distributions of KL → π+π−e+e− with respect to φ. The open histogram shows

generated signals with input asymmetry of 0.147, before any detector, trigger or analysis cuts. The

hatched histogram is the generated events which has at least one electron with momentum less

than 2 GeV/c in the lab frame. In our analysis, we lost all of the hatched region, and we only

accept the remainder, which have the raw asymmetry of 0.229. This is considered to be the origin

of the enhancement in the raw asymmetry.

96

1. Make φ distribution of the signal, and divide it into 32 bins.

2. Evaluate acceptances for each bin, by comparing the number of generated events and accepted

events after all the analysis cuts using the Monte Carlo simulation.

3. Make an acceptance correction to the φ distribution from data bin by bin by using the

acceptances calculated in 2.

4. Calculate the acceptance corrected asymmetry from the acceptance corrected φ distribution.

In this method, the acceptance was averaged over other kinematical variables in each φ bin. Al-

though the raw asymmetry also depends on the other kinematic parameters, the averaged accep-

tance works well since other kinematic parameter distributions are well reproduced by the Monte

Carlo simulation as shown in Figure 6.5.

φ(rad)

Acc

epta

nce

0

0.0025

0.005

0.0075

0.01

0.0125

0.015

0.0175

0.02

0.0225

0.025

-3 -2 -1 0 1 2 3

Figure 7.7: Signal acceptance as a function of

φ. In this plot, we can observe a small shift of

peaks and valleys from the origin. This is orig-

inated from the inefficiency in the low momen-

tum electron, as described in Section 7.1.3.

φ(rad)

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

-3 -2 -1 0 1 2 3

Figure 7.8: The φ distribution of acceptance

corrected data. Dots represent the acceptance

corrected asymmetry from data, while the his-

togram shows the the Monte Carlo events with

input asymmetry of 0.147.

Figure 7.7 shows the acceptance as a function of φ in the Monte Carlo simulation. In the plot, we

can observe a small shift in peaks and valleys from π/2×n. This originated from the inefficiency in

the low momentum electron, as described in Section 7.1.3. With this set of acceptances, we applied

the acceptance correction to the φ distribution of the signal events. The result of this acceptance

correction is shown in Figure 7.8. Dots show the number of events after the acceptance correction,

while the histogram shows the Monte Carlo distribution with the input asymmetry of 0.147. The

φ distribution from the Monte Carlo agree well with that of data. We obtained the acceptance

corrected asymmetry of 0.127± 0.029.

97

We can cross check this value by using the linear relation between input asymmetry and raw

asymmetry in Section 7.1.3. By dividing the raw asymmetry by the linear slope of 1.75, we ob-

tained 0.135± 0.029. Since these two estimations are consistent to each other, we certify that the

acceptance correction is a proper procedure to extract the acceptance corrected asymmetry.

Therefore, we conclude that the acceptance corrected asymmetry is measured to be 0.127±0.029.

As a consistency check, we divide the signal into several event sets. In this study, we used two

neutral beams of +X(west) side and -X(east) side. In addition, we took data in ‘97 winter(Jan.∼Mar.)

and ‘97 Summer(July∼Aug.). This makes four different datasets, Winter/West, Winter/East,

Summer/West, and Summer/East. We looked at these data samples to examine the beam and run

period dependency. Table 7.3 shows the results of the asymmetries in KL → π+π−e+e− of four

datasets. Both raw and acceptance corrected asymmetries show no significant divergence from the

nominal value in each data sample.

Table 7.3: Asymmetries of various data setsData Set Signal Raw Asym. Acc. Corr. Asym.

Winter/West 348 0.172± 0.053 0.069± 0.053

Winter/East 360 0.261± 0.051 0.165± 0.052

Summer/West 241 0.311± 0.061 0.200± 0.063

Summer/East 224 0.161± 0.066 0.062± 0.067

West 589 0.227± 0.040 0.123± 0.041

East 584 0.223± 0.040 0.125± 0.041

Winter 708 0.216± 0.037 0.117± 0.037

Summer 465 0.239± 0.045 0.135± 0.046

Total 1173 0.237± 0.029 0.127± 0.029

7.3 Systematic Uncertainty

Generally, the source of the systematic uncertainty in the asymmetry measurement is common to

the form factor measurement. We will follow the evaluation of systematic uncertainty in the form

factor measurement here. As we saw in Section 6.4, the acceptance correction could be affected

from the imperfectness of the Monte Carlo simulation. Therefore, we will discuss the evaluation

of the uncertainty for both raw and acceptance corrected asymmetry here.

7.3.1 Smearing by Detector Resolution

The smearing by the detector resolution was critical for the asymmetry measurement. The asym-

metry can be diluted by the following mechanisms.

98

The first mechanism is a resolution effect at the sin φ cosφ ∼ 0. Around such a critical region,

the ambiguity from a finite detector resolution easily gives the event transition from the positive

to negative sin φ cosφ, or vice versa. This kind of event swap possibly dilutes the raw asymmetry.

This situation is shown in Figure 7.9. The plot shows the sinφ cos φ distribution from the Monte

Carlo simulation of KL → π+π−e+e−. The hatched region around the origin shows the wrong sign

events; the reconstructed sin φ cosφ had a different sign of real sinφ cos φ. Figure 7.10 shows the

distribution of ∆φ between real φ and reconstructed φ for wrong sign events. The ∆φ plot shows

a sharp peak at origin and suggests that the dilution is caused by the resolution effect.

Second mechanism is X track swapping at the track reconstruction. As shown in Figure 7.11,

two close X tracks can be swapped when connecting upstream and downstream tracks at the

analysis magnet. This track swapping kept the momenta and flight directions of the charged

particles in the X view, while the reconstructed charges were exchanged. This also causes wrong

sign events. This wrong reconstruction gives a striking effect for events with a close electron-

positron pair of Mee < 2MeV/c2, as shown in Figure 7.12. The hatched histogram in Figure 7.12

shows the distribution of wrong sign events in the KL → π+π−e+e− Monte Carlo events with

Mee < 2MeV/c2. Since the wrong sign events are caused by the X track swapping, the wrong

sing events distributed uniformly. We also plot the ∆φ between the real and reconstructed φ, in

Figure 7.13. The wide distribution, in good contrast to the distribution in Figure 7.10, suggests a

poor event reconstruction accuracy for events with Mee < 2MeV/c2. The X track swapping also

appeared as a small uniform distribution of wrong sign events in the dataset with nominal cuts of

Mee > 2MeV/c2, as shown in Figure 7.9.

These effects described above are mostly taken care of by using the detector simulation with

a proper resolution effect in the acceptance correction. The only issue about the smearing is

if the drift chamber resolution was well-understood or not. We used two configurations of the

Monte Carlo simulation for the acceptance correction to evaluate these uncertainties: the Monte

Carlo events were generated with the perfect resolution and 1.5 times worse resolutions for the

drift chambers. The acceptance correction with the perfect chamber resolution gave a reltive

difference of 8.5% to the acceptance corrected asymmetry, and a difference of 2.3% for the worse

case. Therefore, we included 8.7% for the systematic uncertainty to be conservative.

7.3.2 gM1 Form Factor

The uncertainty from the form factor of M1 direct emission was also considered. The ambiguity

from this source could affect the acceptance correction by changing the distributions of kinematic

variables.

This form factor has been measured in Chapter 6 to be a1/a2 = −0.684+0.031−0.043(stat.)±0.053(syst.),

and a1 = 1.05± 0.14(stat.) ± 0.18(syst.). Recently, the measurement of this form factor has been

improved by K → π+π−γ analysis at KTeV, which obtained a1/a2 = −0.729 ± 0.026(stat.) ±0.015(syst.) (preliminary) [50]. Although the obtained values from both K → π+π−γ and K →π+π−e+e− analyses are consistent, we took the difference between those center values as the

99

2 sin φ cos φ

0

1000

2000

3000

4000

5000

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Figure 7.9: The resolution effect causing

wrong sign events. The plot shows the

sin φ cosφ distribution from the Monte Carlo

simulation of KL → π+π−e+e−. The

hatched and solid region around origin shows

the wrong sign events; the reconstructed

sin φ cosφ had a different sign from the real

sin φ cosφ.

φ(gen)-φ(rec)

0

20

40

60

80

100

0 0.5 1 1.5 2 2.5 3

Figure 7.10: The ∆φ (difference between real

φ and reconstructed φ) distribution. The ∆φ

distribution shows a sharp peak at origin and

suggests that the dilution is caused by the res-

olution effect.

100

a)

b)

Analysis Magnet

Figure 7.11: X track swapping mechanism. a) a track reconstruction in X view. b) an alternative

track reconstruction. The reconstructions a) and b) kept the momenta and flight directions of the

charged particles in the X view, while the reconstructed charges were exchanged.

uncertainty.

To evaluate the deviation from the uncertainty, the acceptance correction was done with the

Monte Carlo generated with a1/a2 = -0.73, which was measured by K → π+π−γ analysis. The

asymmetry was changed from 0.127 to 0.124, so the relative difference of 2.4%, was included as

the systematic uncertainty.

7.3.3 Vertexing Quality

The understanding of vertexing quality can be important for the asymmetry measurement, since

the acceptance of a certain decay topology might be changed by the vertexing.

In order to evaluate the uncertainty, we also used the event weighting method with KL →π+π−π0

D events, introduced at Section 6.4.3. This changed the reconstructed asymmetry of the

25000 signal Monte Carlo by 0.8 %. Therefore, we assigned 0.8% as the systematic uncertainty

from this source.

7.3.4 Drift Chamber Inefficiency

As mentioned in Section 6.4.4, there was an inefficiency in the drift chambers around the neutral

beam region. This partly changed the acceptance of detector-dependent event topology. We added

an artificial inefficiency for DC1 and DC2 to be 5% total and obtained 25000 Monte Carlo signal

events. The impact on the reconstructed asymmetry was less than 0.1%, so we concluded that the

chamber inefficiency did not affect the asymmetry.

101

2 sin φ cos φ

0

100

200

300

400

500

600

700

800

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Figure 7.12: The resolution effect causing

wrong sign events with Mee < 2MeV/c2. The

open histogram shows the sin φ cosφ distri-

bution from the signal Monte Carlo. The

hatched and solid histogram indicates the dis-

tribution of wrong sign events. The wrong

sign events distribute uniformly and it was

caused by the X track swapping.

φ(gen)-φ(rec)

0

5

10

15

20

25

30

35

40

45

0 0.5 1 1.5 2 2.5 3

Figure 7.13: The difference between real φ

and reconstructed φ for events with Mee <

2MeV/c2. The wide distribution suggests

that it was not caused by the simple detec-

tor smearing but the track swapping effect.

102

7.3.5 Other Physics Parameters

Table 7.4: Experimental Input Parameter.

Parameter Experimental Input Description

gE1 0.038 E1 Coupling constant

gP 0.15 Charge Radius effect

Experimental input parameters other than the gM1 form factor are listed in Table 7.4. In

the generation of the Monte Carlo events for the acceptance correction, these parameters were

changed to zero to check the stability of reconstructed asymmetry. This is because there is no

decent experimental and theoretical measurements of gP and gE1, as described in Section 6.4.6.

The deviation of the acceptance corrected asymmetry was 6.1% for gE1 = 0.0 and 3.2 % for

gP = 0.0. Therefore, the combined error of 6.9% was included in the systematic uncertainty.

7.3.6 Background Subtraction

Background in the signal region is estimated to be 11.3 ± 3.5 events. Instead of considering

background subtraction, we assumed that all 11 events were in the positive region of the sinφ cosφ

distribution. This gave a difference of 0.9 % of the acceptance corrected asymmetry. We considered

it as the systematic uncertainty.

7.3.7 Analysis Dependency

We examined cut dependencies of the asymmetry just as in the form factor measurement. The

reconstructed asymmetry was very stable against the cut position so that the deviations was

relatively small compared with the statistical uncertainty. We included the deviation of 4.7% as

the systematic uncertainty.

Table 7.5: Deviations of analysis cut dependencies in the acceptance corrected asymmetry.

Cut Variation Description

Z vertex 1.6% 110m < Z < 154m

Cluster Threshold 0.8% Shift HCC threshold by 5 – 10%

Mee cut 0.8% Mee > 2.0 – 4.0 MeV/c2

Mππee cut 0.8% ±6.0 – ±14.0MeV/c2

Pp0kine cut 1.6% −0.0025 – −0.075MeV2/c2

E/p cut 2.3% ±0.06 – ±0.10

p2t cut 3.1% p2

t < 30 – 60MeV2/c2

Total 4.7%

103

7.3.8 Systematic Uncertainty: Summary

The systematic uncertainties in the asymmetry measurement is summarized in Table 7.6.

Table 7.6: Summary of systematic uncertainties.

Source Uncertainty(Absolute uncertainty)

Detector resolution 8.7%(0.011)

M1 form factor 2.4%(0.003)

Vertex quality 0.8% (0.001)

DC ineff. <0.1%(0.000)

Input parameter 6.9% (0.007)

Background subtraction 0.9% (0.001)

Analysis dependency 4.7% (0.005)

Total 12.4% (0.016)

7.4 Asymmetry Measurement: Summary

The angular asymmetry measured in KL → π+π−e+e− is

Asym. =

∫ π/2

0dΓdφdφ − ∫ π

π/2dΓdφdφ∫ π/2

0dΓdφdφ +

∫ π

π/2dΓdφdφ

= 0.127± 0.029(stat.)± 0.016(syst.), (7.5)

good agreement with the theoretical prediction

Asym. ≈ 0.14 (7.6)

in Ref. [22, 23]. An alternative way to calculate the reconstructed asymmetry which utilizes the

linearity between raw asymmetry and input asymmetry gave 0.135 ± 0.029(stat.). This is also

consistent with the experimental result and the theoretical prediction.

104

Chapter 8

Branching Ratio Measurement

In this chapter, we describe how we measured the branching ratio of KL → π+π−e+e−. Unlike

the form factor and asymmetry measurements, background contribution to the branching ratio can

be easily understood since we only have to know the number of background events in the signal

region. Therefore, we here try to relax cuts and enhance the number of signal events in order to

make the statistical error lower, although it will increase the number of background events.

First, we briefly describe the event selection along with above strategy and background esti-

mation. Next, studies of systematic uncertainty will be explained. Finally, we summarize this

branching ratio measurement.

8.1 Event Selection and Background Estimation

In order to increase the number of signal events, we chose a set of constraints of p2t < 0.0001GeV2/c2

and no pp0kine cut, from the signal to background ratio matrix table(Table 5.1). The other

constraints, such as Mee cut, were kept the same as the asymmetry study. This was actually

looser than the original cuts used in the form factor and asymmetry measurements which were

p2t < 0.00004GeV2/c2 and pp0kine < −0.0025GeV2/c2. With this configuration, we obtained

1731 ± 41.6 events in the signal region of 0.493 < Mππee < 0.503GeV/c2, including about 170

background events expected from KL → π+π−π0D.

Figure 8.1 shows the invariant mass distribution of the data and expected background with

the Monte Carlo simulations in the vicinity of the signal region. The background Monte Carlo

simulations were normalized by the KL flux calculated in the Section 5.6 and generated three

times more statistics than the KL flux. We can see that the background is well-understood in

both lower and higher mass regions of the distribution. Dominant background was from the decay

KL → π+π−π0D, and there was some contamination from other sources. Number of expected

background event is 255.4±13.7, as shown in Table 8.1. As a consistency check, we looked at the p 2t

sideband, 0.0001 < p2t < 0.0002GeV2/c2, instead of applying the nominal cut, p2

t < 0.0001GeV2/c2,

105

as shown in Figure 8.2. The Monte Carlo expectation also agrees well with data.

Thus, the number of signal events above background was 1475.6±46.1(stat.), where the statis-

tical errors were summed quadratically. We show the data plot after the background subtraction

in Figure 8.3. A clear peak indicates the signal. The Monte Carlo simulation agrees well with data

in the whole region.

The signal acceptance was also calculated with the Monte Carlo simulation, as shown in Ta-

ble 8.2. The acceptance increased from 1.392%(Table 5.2) to 1.775 % by relaxing the p2t and

Pp0kine cuts.

0

250

500

750

1000

1250

1500

1750

2000

2250

0.46 0.48 0.5 0.52 0.54

Invariant Mass (GeV)

Eve

nts/

0.00

1(G

eV)

0

50

100

150

200

250

300

350

400

450

500

0.48 0.49 0.5 0.51 0.52

Figure 8.1: Invariant mass distribution after final cuts. The upper-right plot is an enlargement of

the signal region. The backgrounds estimated with the Monte Carlo simulations were normalized to

the KL flux. The residual background in the signal region was mostly from KL → π+π−π0D. In this

plot, number of events in the signal region was 1731, while the number of estimated background

events were approximately 255. Thus, the number of expected signal events after background

subtraction was 1475.6.

106

Invariant Mass (GeV)

Eve

nts/

0.00

1(G

eV)

0

250

500

750

1000

1250

1500

1750

2000

2250

0.46 0.48 0.5 0.52 0.54

Figure 8.2: The p2t sideband. Invariant mass distribution after final cuts with 0.0001 < pt <

0.0002GeV2/c2. The distributions of the Monte Carlo simulations were normalized to the KL flux.

The whole distribution was dominated by KL → π+π−π0D.

Table 8.1: The number of background events from each background source, estimated with the

Monte Carlo after final cut. The number of estimated background events were normalized to the

KL flux, so the sum of them gives the number of final background events in the signal region.

Background source # Background events

KL → π+π−π0D 169.2± 13.0 events

KL → π+π−π0 46.7± 2.0

KL → π+π−π0DD 32.2± 3.2

KL → π+π−γ 3.3 ± 0.6

Ke3 double decay 3.0 ± 1.7

Ξ → Λπ0D 0.99± 0.14

Total 255.4± 13.7

107

Invariant Mass (GeV)

Eve

nts/

0.00

1(G

eV)

-100

0

100

200

300

400

0.46 0.48 0.5 0.52 0.54

Figure 8.3: Data and Monte Carlo comparison after background subtraction, estimated with Ta-

ble 8.1 and Figure 8.1. Dots are data, and a histogram is KL → π+π−e+e− Monte Carlo with area

normalization within the signal region, 0.493 < Mπ+π−e+e− < 0.503GeV/c2. In the whole region,

distributions of both data and Monte Carlo are in good agreement.

Table 8.2: The estimated signal acceptance at each analysis phase. The estimation was done by

KL → π+π−e+e− Monte Carlo.

Analysis phase Efficiency

Level 1 trigger 12.8%

Level 2 trigger 9.08%

Level 3 trigger 8.40%

Basic Cuts 2.18%

0.493 < MK < 0.503GeV/c2 2.07%

Pππee < 200GeV/c 2.06%

Vertex χ2 <50 2.05%

Mee > 0.002GeV/c2 1.78%

p2t <0.00010 (GeV2/c2) 1.775%

108

8.2 Systematic Uncertainty

Sources of systematic uncertainties are mostly common to the form factor and asymmetry measure-

ments. Possible sources of uncertainties described here are, detector simulation, analysis constraint,

normalization, and the decay modeling of KL → π+π−e+e−.

For the uncertainty from the normalization, we have already estimated it from the source of

the detector response, KL → π+π−π0D decay modeling and the branching ratio in Section 5.6.

However, most of the uncertainties from the detector response would be canceled out by taking

the ratio of acceptances between signal and normalization modes, since the uncertainty affected

both modes similarly. Therefore, we will redo the systematic uncertainty study for the detector

response here, as well as the KL → π+π−e+e− specific uncertainties such as the a1/a2 form factor

uncertainty.

The major systematic uncertainties on the branching ratio measurement discussed here are as

follows.

• Drift chamber inefficiency in the beam region.

• Physics parameters such as a1/a2.

• Normalization ambiguity.

• Monte Carlo statistics.

• Background subtraction.

After the explanation of each issue, we will summarize the uncertainty study.

8.2.1 Drift Chamber Inefficiency

The drift chamber inefficiency in the beam region was a major systematic uncertainty in the

normalization. However, since the inefficiency was also applied on the signal mode in the branching

ratio measurement, this effect almost canceled out. In fact, although adding the inefficiency of 5 %

in the beam region in DC1 and DC2 changed the KL flux measured with KL → π+π−π0D by 1.9%,

it changed the branching ratio of KL → π+π−e+e− only by 0.17%. Therefore, we determined the

uncertainty from drift chamber inefficiency to be 0.17%.

8.2.2 Vertexing Quality

As shown in Figure 6.10, the understanding of vertex χ2 distribution might affect the branching

ratio measurement. However, this effect is also canceled when taking the acceptance ratio of KL →π+π−e+e− to KL → π+π−π0

D. The uncertainty from this source was extracted with weighting

the Monte Carlo events, same as the other measurements. The modified Monte Carlo gave the

109

difference of 0.87% on the branching ratio. Therefore, we assigned 0.87 % as the uncertainty from

the understanding of vertex quality.

8.2.3 Normalization Ambiguity

The uncertainty from the flux normalization with KL → π+π−π0D is 3.15 %, which is dominated by

the uncertainty from the branching ratio measurement of KL → π+π−π0D of 3.14 %, as explained in

Section 5.6. In order to classify the sources of systematic uncertainties into inside and outside of the

analysis, we separate the uncertainty in the branching ratio measurement of KL → π+π−π0D and

other small contributions. therefore, the total uncertainty from the normalization was determined

as [3.14(external)⊕ 0.33(internal)]% from the normalization study.

8.2.4 KL → π+π−e+e− Monte Carlo Generation

The uncertainties related to the Monte Carlo simulation is explained here. One possible uncertainty

source is physics input parameters in the matrix element, such as a1/a2. This kind of uncertainty

was evaluated by varying the parameter within the probable region. The other uncertainty is the

statistical error in the signal Monte Carlo.

Monte Carlo Statistics

In order to evaluate the acceptance, we generated 9.0 millions KL → π+π−e+e− Monte Carlo

events and we accepted 0.16 million events after all detector simulation and the analysis cuts. This

caused the systematic uncertainty of 0.25%.

Parameters in the Matrix Element

We also introduced the form factor for the acceptance calculation. Since fitted form factors a1/a2

and a1 show a strong correlation between them, as shown in Figure 6.4, we checked variations

of the signal acceptance along the one standard deviation contour of these values. The largest

difference was 1.06%, so we included it as the uncertainty due to a1/a2 and a1.

Since the other physical input parameters in the matrix element are also defined with some un-

certainties, we have to take these uncertainties into account. The variations of physics parameters

and the uncertainties are shown in Table 8.3. We quadratically summed these errors and obtained

an error of 2.60%. Therefore, we assigned 2.6% as a systematic uncertainty due to the physics

parameters.

8.2.5 Background Subtraction

Here, we examine the background-related uncertainties.

110

Table 8.3: Stability to parameter fluctuations. Expected error from the Monte Carlo statistics was

0.28%.Parameter variation Acceptance Uncertainty

a1/a2, a1 1σ contour 1.06%

η+− + 1σ -0.29%

η+− − 1σ +0.63%

Φ+− + 1σ -0.24%

Φ+− − 1σ -0.79%

gE1 = 0.0 -0.39%

gP = 0.0 -2.11%

Total 2.60%

Statistical error

Major background sources and expected contaminations are listed in Table 8.1. The background

was mostly from KL → π+π−π0D, and there was a small contribution of conversion backgrounds

from KL → π+π−π0. The uncertainty from statistic fluctuation of the background is estimated as

13.7 events, corresponding to 0.93% of uncertainty in the signal.

Sideband comparison

In order to confirm the expected number of background events, we compared the Monte Carlo

expectations with the data in the sideband of the signal. As shown in Figure 8.4, the estimated

background level agreed with the data. The uncertainty was taken from the largest discrepancy,

7.37%, between data and the Monte Carlo estimation.

We determined that the uncertainty in the number of background events was 18.8, which is

7.37% of 255.4 background events. This corresponds to 1.27 % of uncertainty in the branching

ratio measurement. Thus, we included 1.27 % as the systematic uncertainty.

Miscellaneous Sources

The branching ratio measurements in other background events are also related to the uncertainty.

The errors were taken from PDG [9]. This gives total 0.43% of the uncertainty in the branching

ratio measurement.

Summary

The uncertainty from the background estimation is, thus, 1.6 % in the branching ratio measure-

ment.

111

Data1760 ± 42.0

MC1861.9 ± 36.3

Data1731 ± 41.6

MC1731 ± 20.3

Data45 ± 6.7

MC42.4 ± 3.7

Data734 ± 27.1

MC787.1 ± 25.0

Data81 ± 9.0

MC87.0 ± 5.8

Data25 ± 5.0

MC20.5 ± 3.7

Inv. Mass(GeV)

p t2 (G

eV2 /c

2 )

0

0.05

0.1

0.15

0.2

0.25

0.3

x 10-3

0.48 0.49 0.5 0.51 0.52

Figure 8.4: Comparison between data and the Monte Carlo estimation in the sideband. The Monte

Carlo was normalized to the KL flux and included the signal Monte Carlo. The largest discrepancy

was 7.37% in this plot.

112

8.2.6 Analysis Dependency

As in the form factor and asymmetry analysis, consistency check in the analysis dependency with

various cuts was examined here. The results are shown in Table 8.4. The total uncertainty, 0.7%

was assigned to the systematic uncertainty.

Table 8.4: Summary of analysis dependency.

Cut Normalization BR measurement Estimation Method

Z vertex 0.09% 0.2% 110m < Z < 154m

Cluster Threshold < 0.1% 0.1% Shift HCC threshold

Mee cut 0.07% < 0.1% Mee > 0.004, 0.008GeV/c2

Min. Cluster Sep. 0.3% 0.3% Min. Cluster Sep. > 20 cm

Meeγ cut 0.4% 0.4% 0.130 < Meeγ < 0.136GeV/c2

E/p cut 1.8% 0.3% ±0.06 – ±0.10

p2t cut 0.6% 0.4% p2

t < 60 – 100MeV2/c2

Total 0.7 %

8.2.7 External Systematic Sources

So far, we have considered the systematic uncertainties originated from both analysis-induced(internal)

and external sources. To clarify the responsibilities of those uncertainties, we split off the external

systematic uncertainty here. We classified the branching ratio of KL → π+π−π0D(3.14%) and the

input parameters for KL → π+π−e+e− Monte Carlo(2.60%) as the external uncertainties.

8.2.8 Systematic Uncertainty: Summary

We summarize the systematic uncertainty in the branching ratio calculation in Table 8.5. The

largest uncertainty was the error in the branching ratio of KL → π+π−π0D. The external systematic

uncertainty is split off here.

8.3 Branching Ratio Measurement: Summary

We determined the branching ratio for the decay KL → π+π−e+e−, based on 1475.6 signal events

as:

BR(KL → π+π−e+e−) = (3.55± 0.11(stat.)± 0.07(syst.internal) ± 0.14(syst.external)) × 10−7.

The ratio between the signal and the normalization modes, which is not affected by the error

on the branching ratio measurement of KL → π+π−π0D, is:

BR(KL → π+π−e+e−)BR(KL → π+π−π0

D)= (2.36± 0.07(stat.)± 0.05(syst.internal) ± 0.06(syst.external)) × 10−4.

113

Table 8.5: Summary of uncertainty in branching ratio measurement.

Source Acceptance Uncertainty

Statistical error 3.13%

BR(KL → π+π−π0D) 3.14 %

MC KL → π+π−e+e− input param. 2.60%

Normalization; other sources 0.33%

Detector(Drift chamber Ineff.) 0.17%

Vertex Quality 0.87%

KL → π+π−e+e− MC stat. 0.25%

Background-originated 1.6 %

Analysis dependency 0.7 %

Total (3.13(stat.) ⊕ 2.00(syst.internal) ⊕ 4.08(syst.external))%

114

Chapter 9

Discussion

So far, we derived physics parameters from KL → π+π−e+e− data sample. In this chapter, using

the results, we discuss the implications of the results.

9.1 Form Factor Measurement

The M1 direct emission(DE) form factor was determined as

a1/a2 = −0.684+0.031−0.043(stat.) ± 0.053(syst.) (9.1)

a1 = 1.05± 0.14(stat.)± 0.18(syst.). (9.2)

This result compares well with a preliminary result of KL → π+π−γ performed by the KTeV

collaboration [31],

a1/a2 = −0.729± 0.026(stat.)± 0.015(syst.). (9.3)

From the theoretical point of view, a1 and a2 depend on the mixing angle, θη−η′ , between the

SU(3) nonet members η and η′, and the flavor SU(3) breaking parameter, ξ.

Figure 9.1 shows our result and results on a1/a2 direct measurements from the KTeV and

previous(Fermilab E731)*1 experiment in the KL → π+π−γ [31, 32]. The direct measurements

with KL → π+π−γ were made by directly fitting the Eγ spectrum in the center-of-mass frame.

We obtained consistent results of a1/a2 ∼ −0.7.

Moreover, the form factor a1/a2 can be also extracted from the DE branching ratio measurement

through θη−η′ determination in the chiral Lagrangian framework by Lin and Valencia [29, 30]. Their

model predicts the mixing angle to be θη−η′ = −20 from the experimental DE branching ratio,

BR(KL → π+π−γ)DE ∼ 3 × 10−5, and the SU(3) breaking parameter, ξ = 0.17, as shown in

*1The Eγ spectrum from E731 data was reanalyzed and directly fitted [32], and the fit result is different from

the published result of a1/a2 = −1.8 ± 0.2 [37]. This is because the published result assumed the Lin and Valencia

model to calculate the form factor with the DE branching ratio.

115

Figure 9.2 [37, 38]. The form factor can then be evaluated with the obtained mixing angle and

the SU(3) breaking parameter from a relation shown in Figure 9.1. This gives the form factor of

a1/a2 = −1.8 ± 0.2.

The predicted value, a1/a2 = −1.8 ± 0.2, does not agree well with the direct measurement

results, a1/a2 ∼ −0.7. Conversely, this means that in our configuration of a1/a2 ∼ −0.7, the

Lin and Valencia model predicts the DE branching ratio of BR(KL → π+π−γ) < 1.0 × 10−5, in

contradiction to the recent experimental result, BR(KL → π+π−γ) = (3.19± 0.09) × 10−5 [31].

Therefore, our results related to the θη−η′ mixing angle and the SU(3) breaking parameter does

not support the Lin and Valencia model.

Figure 9.1: Results on the DE form fac-

tor, a1/a2. The horizontal axis means θη−η′ ,

and the vertecal axis means the direct a1/a2

measurement results from KTeV [31] and

E731(a1/a2 ∼ −0.7) [32], and the model-

dependent result from E731. The model-

dependent result(a1/a2 = −1.8 ± 0.2) was

extracted from the SU(3) breaking param-

eter, ξ = 0.17, and θη−η′ = −20 ± 1,

which was calculated with the DE branching

ratio(3× 10−5) by using the Lin and Valencia

model [29] shown in Figure 9.2.

Figure 9.2: Relation between θη−η′ and the

DE branching ratio. The mixing parameter

θη−η′ is related to the DE branching ratio

in the Lin and Valencia model [29], which

predicts θ ∼ −20 ± 1 from E731 result,

BR(KL → π+π−γ)DE = (3.19 ± 0.16) ×10−5 [37]. I is for ξ = 0.00(non-breaking

limit), and II is for ξ = 0.17, the reasonable

value referred in Ref. [30]. In the Lin and Va-

lencia model, the a1/a2 was extracted from

θ ∼ −20± 1 and ξ = 0.17 as compared with

Figure 9.1.

116

9.2 Asymmetry Measurement

We have measured the angular asymmetry in KL → π+π−e+e− with a good precision:

Asym. =

∫ π/2

0dΓdφdφ − ∫ π

π/2dΓdφdφ∫ π/2

0dΓdφdφ +

∫ π

π/2dΓdφdφ

= 0.127± 0.029(stat.)± 0.016(syst.).

We here discuss any possible contributions to the angular asymmetry.

One possible contribution to the angular asymmetry is the final state interaction. However,

this contribution cancels within the highest order of the electromagnetic interaction, since a) an

interaction between π+ and π−(e+ and e−) does not contribute to the angular asymmetry, and

b) the angle φ could be shifted by the electromagnetic interaction between π+ and e+, but they

are distorted symmetrically around φ = 0. Therefore, the final state interaction does not affect

the angular asymmetry. A theoretical calculation with the chiral perturbation framework supports

this description [26]. Therefore, we conclude that the contribution from the final state interaction

is negligible.

At one point, there was an issue on the angular asymmetry, whether this indicated the T

violation effect or not. While reversing the momenta of the final state particles change the sign of

φ(T-odd), the initial and final states in the decay have not been interchanged. Therefore, strictly

speaking, this observation is not directly connected to T violating interaction.

However, if the CPT theorem would not hold, we could consider a different scenario in T and

CPT violation. Recent discussions suggest that CPT violation may be responsible for the whole

angular asymmetry in KL → π+π−e+e− [41, 42]. This assumption needs an unnatural parameter

configuration, but we cannot reject this possibility at this time.

The orthodox interpretation of the asymmetry in KL → π+π−e+e− is that CPT invariance

holds, but the CP violating effect is responsible for the whole angular asymmetry. In this context,

the theoretical prediction agrees well with the experimental result. In addition, the asymmetry is

extracted only with natural and well-known CP violating parameters. This is the most preferable

interpretation which is consistent with the current knowledge of CP violation.

9.3 Branching Ratio Measurement

The branching ratio measured in this analysis is

Br(KL → π+π−e+e−) = (3.55± 0.11(stat.) ± 0.07(syst.internal) ± 0.14(syst.external)) × 10−7,(9.4)

which agrees with the previous publication [40],

Br(KL → π+π−e+e−) = (3.2 ± 0.6(stat.) ± 0.4(syst.)) × 10−7. (9.5)

In this analysis, we improved the precision of the branching ratio by four times than of the previous

measurement. The largest systematic uncertainty in the previous analysis was due to the lack of

117

knowledge of gM1 form factor. In our analysis, we have reduced the uncertainty with a better

understanding of the form factor.

9.4 Remarks and Future Prospects

These measurements represent the first step in our understanding of the decay KL → π+π−e+e−.

The operation of fixed-target experiment E799-II continues for 1999 and we plan to collect two

to four times the statistics used in this thesis. In branching ratio measurement using the technique

described here, we have already achieved the systematic limit by the uncertainty in the π0 Dalitz

decay branching ratio. The accuracy of the angular asymmetry and the gM1 form factor can be

improved in 1999 run because our errors are still dominated by statistics.

This mode is also being studied at CERN NA48 and their result is expected to come out soon.

They have reported preliminary results on the branching ratio and asymmetry measurements based

on 458±22 events [54]. The branching ratio were presented as (2.9±0.15)×10−7. The experiment-

dependent(i.e., the raw asymmetry for NA48) was (20 ± 5)% and their Monte Carlo and detector

simulation predicted that the input asymmetry of 14 % was modified to the raw asymmetry of

22 %. Those results are in good agreement with our results.

118

Chapter 10

Conclusion

We measured the CP violating effect in the decay KL → π+π−e+e− with 1162 signal events. The

acceptance corrected asymmetry is

Asym. =

∫ π/2

0dΓdφdφ − ∫ π

π/2dΓdφdφ∫ π/2

0dΓdφdφ +

∫ π

π/2dΓdφdφ

= 0.127± 0.029(stat.)± 0.016(syst.).

This represents a first CP violation appearance in a kinematic parameter and the largest CP

violating effect observed in the world, as well as the fourth appearance of the indirect CP violating

effect.

The branching ratio of KL → π+π−e+e− was also measured with 1475.6 signal events. The

result is

Br(KL → π+π−e+e−) = (3.55± 0.11(stat.) ± 0.07(syst.internal) ± 0.14(syst.external)) × 10−7,

which agrees with the previous publication,

Br(KL → π+π−e+e−) = (3.2 ± 0.6(stat.) ± 0.4(syst.)) × 10−7.

If we split off the uncertainty from the branching ratio measurement in the KL → π+π−π0D, the

ratio of the branching ratios becomes

BR(KL → π+π−e+e−)BR(KL → π+π−π0

D)= (2.36± 0.07(stat.)± 0.05(syst.internal) ± 0.06(syst.external)) × 10−4.

The gM1 direct emission form factor was determined as

a1/a2 = −0.684+0.031−0.043(stat.) ± 0.053(syst.)

a1 = 1.05± 0.14(stat.)± 0.18(syst.).

119

This result compares well with a preliminary result of KL → π+π−γ measured by KTeV collabo-

ration [31],

a1/a2 = −0.729± 0.026(stat.)± 0.015(syst.).

Our results on the gM1 form factor measurements in KL → π+π−e+e− and KL → π+π−e+e−,

do not support the Lin and Valencia model.

In conclusion, this experiment found the largest indirect CP violation effect in a kinematic vari-

able. This result is consistent with any other related measurement and the theoretical prediction.

120

Appendix A

Maximum Likelihood Method

This describes how the form factor of gM1 direct emission was determined with the maximum

likelihood method. This is the explanation to the application for the form factor measurment, and

does not intend to show a general formalism of the maximum likelihood method.

A.1 Definition

Let us consider to determine a set of physics parameters a from measurements of a set of certain

observables x. Assume that we have a collection of N signal events corresponding to the inde-

pendent measurement of variable sets xi, where i runs 1 to N . We wish to obtain the parameter

set a of a fitting function y(xi) = y(xi; a) from these data. For each event, we convert y(xi) to a

normalized probability density function

Pi = P (xi; a) (A.1)

evaluated at the observables xi. The likelihood function is simply multiplying Pi event by event:

L(a) =N∏

i=1

Pi. (A.2)

Equation A.2 shows a probability to realize the sets of observables xi with a configuration of

parameters a. We can consider that these observables would be given with the most possible

values of parameters a. Therefore, the maximum-likelihood value of parameters is obtained by

maximizing L(a) with respect to the parameter a. To maximize Equation A.2, we sometimes take

a logalithm for both sides of the equation, in order to maximize it easy,

logL(a) =N∑

i=1

Pi. (A.3)

In addition, a detection efficiency of the signal is crucial to the solution of the problem. This

is both detector- and observables-dependent. The detection efficiency would give a certain weight

121

to the probability density function Pi. So the probability to observe a single event with a certain

observed variables xi is obtained as

Pi = Norm · Ai · P(xi; a) (A.4)

where the factor Ai is the detection efficiency and the factor P(xi; a) is a probability to decay

a particle with the variables x = xi. In our case, P(xi; a) corresponds to KL → π+π−e+e−

differencial cross section with related to xi. Norm is the normalization factor calculated later.

For the decay of KL → π+π−e+e− at the KTeV detector, the probability density function Pi

with normalization becomes

Pi = Norm · Ai · dΓ(xi; a)dx

(A.5-a)

a = (a1/a2, a1, η+−, Φ+−, gP, gE1) (A.5-b)

xi = (φi, Mππi, Meei, cos θπ+i, cos θe+i) (A.5-c)

where dΓ(xi; a)/dx is a differential cross section of KL → π+π−e+e− with respect to x.

To normalize Pi, we integrate Pi over physically possible value of x.

Norm ·∫

dx(A(x) · dΓ(x; a)dx

) = 1, (A.6)

so the normalization factor for Pi is obtained as

Norm =1∫

dx(A(x) · dΓ(x;a)dx )

. (A.7)

Finally, the probability density function Pi, and the likelihood function L becomes

Pi =Ai · dΓ(xi;a)

dx∫dx(A(x) · dΓ(x;a)

dx ), (A.8)

lnL =∑

i

ln Pi =∑

i

lnAi · dΓ(xi;a)

dx∫dx(A(x) · dΓ(xi;a)

dx ). (A.9)

In following sections, we can assume that the detection efficiency Ai is independent of parameter

a so that Ai can be written as Ai = A(xi).

A.2 Use of Monte Carlo

Since it is impossible to give an exact analytic form of the detection efficiency and the normalization

factor, we here use the Monte Carlo simulation to determine them. In this section, the parameter

a is fixed as a = a0, to make the problem easier.

First, we consider the simplest case, the detection efficiency is unity over any x, i.e. A(x) =

1. When the Monte Carlo simulation is performed according to the differencial cross section

dΓ(x; a0)/dx, a relation to a variable n(x; a0), which is the number of events in an infinitesimal

phase space dx, becomesdΓ(x; a0)

dxdx ∝ n(x; a0)

Ndx (A.10)

122

if the number of generated events, N, goes to infinity.

We have to consider the normalization to derive the equation. Integrating both sides of Equa-

tion A.10, we obtain ∫dx

dΓ(x; a0)dx

= Γ(a0) (A.11)

for the left side and ∫dx

n(x; a0)N

= 1 (A.12)

for the right side. Therefore, Equation A.10 becomes

dΓ(x; a0)dx

dx =n(x; a0)

NΓ(a0)dx (A.13)

Next, we take the detection efficiency into account. The detection efficiency in dx is defined by

the ratio of the number of survivals through all detector simulation and cut to that of generated

events in dx.

A(x) · dΓ(x; a0)dx

dx =n′(x; a0)n(x; a0)

n(x; a0)N

Γ(a0)dx (A.14-a)

=n′(x; a0)

NΓ(a0)dx (A.14-b)

where n′(x; a0) is the number of events in dx after all cut. After integration, we obtain∫dxA(x) · dΓ(x; a0)

dx=

∫dx

n′(x; a0)N

Γ(a0) (A.15-a)

= (Overall Acceptance)(a0) · Γ(a0) (A.15-b)

(Overall Acceptance)(a0) =N ′(a0)

N(A.15-c)

where N ′(a0) is the number of events after all cut.

Finally, we calculate A(xi) for the numerator. Assuming that the acceptance is constant to the

detector resolution,

A(xi) = m′(xi)/m(xi) (A.16)

where m(xi) is the total number of events of Monte Carlo generation with xi and m′(xi) is surviving

events after all the detector simulation and analysis cuts with xi.

We can write the equation A.9 as

lnL =∑

i

lnm′(xi)/m(xi) · dΓ(xi;a0)

dxN ′(a0)

N · Γ(a0)(A.17)

A.3 Reduction of Monte Carlo Generation

We consider here to reduce the Monte Carlo generation against arbitrary a for the economy of

time and processing power. We just want a general form of

dΓ(x; a)dx

dx =n(x; a)

NΓ(a)dx. (A.18)

123

First, we can write the left side of above equation with the factor A(x)

A(x)dΓ(x; a)

dxdx =

dΓ(x; a)/dx

dΓ(x; a0)/dx· A(x) · dΓ(x; a0)

dxdx (A.19)

This lead to

A(x)dΓ(x; a)

dxdx =

dΓ(x; a)/dx

dΓ(x; a0)/dx· n′(x; a0)

NΓ(a0)dx (A.20)

This means that the number of event from general form is given as a weighted number of events

generated with a parameter a0 and weight dΓ(x;a)/dxdΓ(x;a0)/dx .∫

dxA(x) · dΓ(x; a)dx

=∑

j

(n′(xj ; a0)

N· dΓ(xj ; a)/dx

dΓ(xj ; a0)/dx) · Γ(a0) (A.21)

Finally, we can give the general form of the likelihood function as

ln L(a) =∑

i

lnm′(xi)/m(xi) · dΓ(xi;a)

dx∑j(

n′(xj ;a0)N · dΓ(xj;a)/dx

dΓ(xj;a0)/dx ) · Γ(a0)(A.22)

In our study, we do not have to calculate m′(xi)/m(xi) and Γ(a0) since their contributions are

only constants. So we maximized this ln L(a);

ln L(a) =∑

i

lndΓ(xi;a)

dx∑j(

n′(xj ;a0)N · dΓ(xj;a)/dx

dΓ(xj;a0)/dx )+ Const. (A.23)

A.4 Fit Example

In this thesis, we generated 30 million Monte Carlo events of KL → π+π−e+e− with the parameters

of a1/a2 = −0.70 and a1 = 1.0 for the acceptance calculation in the denominator of Equation A.23.

After all cut, we obtained about 0.5 million events of the signal and utilized in the study. The

maximization was done with MIGRAD and MINOS in MINUIT since we obtained analytic form

of the likelihood function. Figure 6.4 shows a result of maximum likelihood fit. The fit results

based on 1173 events of the signal are

a1/a2 = −0.684+0.031−0.043 (A.24)

a1 = 1.05± 0.14 (A.25)

124

Appendix B

Pp0kine: Kinematics with

a Missing Particle

Sometimes one particle in the final state of a decay was either missing or ignored in this experiment.

In KL → π+π−e+e− analysis, the KL → π+π−π0D with the γ missing is the largest background.

Here, we extract the useful kinematical relation to handle the event with a missing or ignored

particle [51, 52, 53].

In general, assuming that a particle of mass M and four momentum P = (E, ~P ) will decay

into a set of particles which can be observed and reconstructed and one particle which cannot.

For a missing particle, we define the mass and momentum as mmis and pmis = (Emis, ~pmis). For

reconstructed particles, we treat them as a single system with a mass mobs and a momentum

pobs = (Eobs, ~pobs). In this chapter, all of the quantities with an asterisk(*) refers to quantities in

the center-of mass-system of the decay particle.

From the invariant mass squared

P · P = M 2 = m2obs + m2

mis + 2(EobsEmis − ~pobs · ~pmis) (B.1)

and

P · pmis = ME∗mis = m2

mis + EobsEmis − ~pobs · ~pmis, (B.2)

we obtain

E∗mis =

M2 − m2obs + m2

mis

2M

p∗2mis =(M2 − m2

obs + m2mis)

2 − 4m2obsm

2mis

4M 2. (B.3)

In a fixed target experiment, the decay particles are boosted along the line spanned between

the target and decay vertex point. In this context, the transverse momentum pt of the observed

particle(s) can determine the magnitude of longitudinal momentum of the observed particle system.

125

This, thus, is equivalent to measure the longitudinal momentum of the unobserved particle in the

decay rest frame.

p∗2obs‖ = p∗2mis‖ = p∗2obs − p2t

=(M2 − m2

obs + m2mis)

2 − 4m2obsm

2mis

4M 2− p2

t

=(M2 − m2

obs + m2mis)

2 − 4m2obsm

2mis − 4M 2p2

t

4M 2. (B.4)

We have obtained the magnitude of the longitudinal component of the momentum in the decay’s

center-of-mass frame.

Historically, a quantity “Pp0kine” has been often used for the analyses with a missing particle

instead of p∗2obs‖. To examine this, consider the product Ppobs

Ppobs = ME∗obs = EEobs − ~P · ~pobs

= EEobs − ~P · ~pobs‖. (B.5)

Regrouping and squaring, we have

(E2obs − |~pobs‖|2)|~P |2 − 2ME∗

obs(~pobs‖ · ~P ) + M2(E2obs − E∗2

obs) = 0. (B.6)

Solving for P, and substituting m2obs + p2

t for E2obs − |~pobs‖|2, we obtain

|~P | = γβM

=(E∗

obs|~pobs‖| ± Eobs|~p∗obs‖|)Mm2

obs + p2t

=|~pobs‖|(M2 − m2

obs + m2mis) ± (|~pobs|2 + m2

obs)1/2[(M2 − m2

obs + m2mis)

2 − 4m2obsm

2mis − 4M 2p2

t ]1/2

2(m2obs + p2

t )(B.7)

When we assume |~pobs‖| = 0 in Equation B.7, we obtain a variable called “Pp0kine”:

Pp0kine =(M2 − m2

obs + m2mis)

2 − 4m2obsm

2mis − 4M 2p2

t

4(m2obs + p2

t ). (B.8)

The quantity “Pp0kine” is connected with a square of a longitudinal momentum shown in

Equation B.4 as follows:

Pp0kine = p∗2obs‖ ·

M2

m2obs + p2

t

. (B.9)

Pp0kine is equivalent to p∗2obs‖ with a rescaling factor, M 2/(m2

obs + p2t ), which boosts to the frame

of |~pobs‖| = 0. This is invariant in the sense that the distribution in Pp0kine is independent of the

decay particle’s momentum.

In this thesis, we refer Pp0kine as the relation in KL → π+π−π0 and a missing particle assigned

to the π0.

126

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